Short Answer
How often are lives saved by civilian private citizens using guns in the United States?
Probably in the mid-hundreds to low single digit thousands of times per year.
But in many cases, it is intrinsically difficult to know if using a gun in self-defense saved a life, so one would like to resort to statistical evidence.
Statistically, firearms are so vastly more likely to be used criminally than to be used in legal self-defense, that the net impact on homicide rates of firearm ownership is unequivocally negative. On a net basis, firearms result in lives lost, rather than lives saved. Having a gun makes you much more likely to be killed and much less safe from violent crime.
As the example of bank robberies illustrates, even when one is not actually shooting someone in a self-defensive use of a firearm, one is usually making it more likely that an innocent person will be killed, not less likely.
Firearms also don't prevent tyranny and generally make civil wars and coups more deadly without changing the ultimate results.
Higher rates of firearm ownership by civilians does, however, reliable and predictably increases the rates of firearm suicides and firearm accidents involving children. Increases in firearm suicides do not materially reduce or increase suicides from other causes.
Long Answer
There is a great deal of research on the question, but there is not really any study, including Kleck's that provide a definitive answer in one place.
Kleck's study is, to put in mildly, garbage. Other research is better, but necessarily tackles bits of this difficult to answer question at a time from many different angles.
Some Key, High Quality Research On The Question
Various publications of the Harvard Injury Control Research Center (available at the link) address the issue. Their results can be summarized as follows:
- Guns are not used millions of times each year in self-defense.
- Most purported self-defense gun uses are gun uses in escalating arguments, and are both socially undesirable and illegal.
- Firearms are used far more often to intimidate than in self-defense.
- Guns in the home are used more often to intimidate intimates than to thwart crime.
- Adolescents are far more likely to be threatened with a gun than to use one in self-defense.
- Criminals who are shot are typically victims of crime.
- Few criminals are shot by law-abiding (non-law enforcement) citizens.
- Self-defense gun use is rare and not more effective at preventing injury than other protective actions.
These studies include:
Hemenway, David. Survey research and self-defense gun use: An explanation of extreme overestimates. Journal of Criminal Law and Criminology. 1997; 87:1430-1445 (the claim of many millions of annual self-defense gun uses by American citizens is invalid).
Hemenway, David. The myth of millions of annual self-defense gun uses: A case study of survey overestimates of rare events. Chance (American Statistical Association). 1997; 10:6-10 (the claim of many millions of annual self-defense gun uses by American citizens is invalid).
Cook, Philip J; Ludwig, Jens; Hemenway, David. The gun debate’s new mythical number: How many defensive uses per year? Journal of Policy Analysis and Management. 1997; 16:463-469 (the claim of many millions of annual self-defense gun uses by American citizens is invalid).
Hemenway, David; Miller, Matthew; Azrael, Deborah. Gun use in the United States: Results from two national surveys. Injury Prevention. 2000; 6:263-267 (criminal court judges who read the self-reported accounts of the purported self-defense gun use rated a majority as being illegal, even assuming that the respondent had a permit to own and to carry a gun, and that the respondent had described the event honestly from his own perspective).
Hemenway, David; Azrael, Deborah. The relative frequency of offensive and defensive gun use: Results of a national survey. Violence and Victims. 2000; 15:257-272 (firearms are used far more often to frighten and intimidate than they are used in self-defense).
Azrael, Deborah R; Hemenway, David. In the safety of your own home: Results from a national survey of gun use at home. Social Science and Medicine. 2000; 50:285-91 (guns in the home are used more often to frighten intimates than to thwart crime; other weapons are far more commonly used against intruders than are guns).
Hemenway, David; Miller, Matthew. Gun threats against and self-defense gun use by California adolescents. Archives of Pediatrics and Adolescent Medicine. 2004; 158:395-400 (young people were far more likely to be threatened with a gun than to use a gun in self-defense, and most of the reported self-defense gun uses were hostile interactions between armed adolescents).
May, John P; Hemenway, David. Oen, Roger; Pitts, Khalid R. When criminals are shot: A survey of Washington DC jail detainees. Medscape General Medicine. 2000; June 28. www.medscape.com (one in four detainees in a D.C. jail had been wounded by gunfire, in events that appear unrelated to their incarceration. Most were shot when they were victims of robberies, assaults and crossfires. Virtually none report being wounded by a “law-abiding citizen.”)
May, John P; Hemenway, David. Oen, Roger; Pitts, Khalid R. Medical Care Solicitation by Criminals with Gunshot Wound Injuries: A Survey of Washington DC Jail Detainees. Journal of Trauma. 2000; 48:130-132 (virtually all criminals who have been shot went to the hospital, and can describe in detail what happened there).
May, John P; Hemenway, David. Do Criminals Go to the Hospital When They are Shot? Injury Prevention. 2002; 8:236-238 (virtually all criminals who have been shot went to the hospital, and can describe in detail what happened there).
Hemenway D, Solnick SJ. The epidemiology of self-defense gun use: Evidence from the National Crime Victimization Surveys 2007-2011. Preventive Medicine. 2015; 79: 22-27 (victims use guns in less than 1% of contact crimes, and women never use guns to protect themselves against sexual assault (in more than 300 cases)).
A Better Designed Survey Shows Kleck Grossly Overestimated Defensive Uses Of Firearms
There are on the order of 60,000-120,000 reported defensive uses of firearm per year according to the National Crime Victimization Survey. This places an upper bound on the number of lives saved by such defensive uses of firearms.
McDowall DB, Wiersema B., "National Archives of Criminal Justice Data. National Crime Victimization Survey: 1986–1991, 1992–1995." http:\www.icpsr.umich.edu:80/NACJD. (the National Crime Victimization Survey estimates that each year about one million violent crimes involve guns).
The incidence of civilian defensive firearm use by US crime victims. Am J Public Health 1994;84:1982–5 (victims use guns in self defense perhaps 60,000 to 120,000 times each year).
The number of lives saved by guns is necessarily significantly smaller than this 60,000 to 120,000 number.
Less than half of these self-defense uses of firearms are in cases involving violent crime as opposed to property crime, and involve violent crimes where death was a likely outcome for the victim.
The most difficult to estimate question is what proportion of cases where a gun is brandished or a warning shot is fired, without actually shooting someone, saves a life, since the vast majority of defensive use of firearms do not involve actually shooting someone.
More Than Half Of People Who Are Shot Die
About 9,200 people are non-fatally shot with firearms each year (a five year average from 2007-2011), and about 11,000 people were fatally shot in violent crimes in 2011, according to the U.S. Department of Justice.
Justified Homicides By Private Citizens Are Rare
Criminal uses of firearms are one or two orders of magnitude more common than justified uses of firearms.
In the time period from 2007 to 2021 there were between 257 and 414 justified homicides by private citizens (as opposed to law enforcement officers) each year.
This establishes something close to a lower bound of lives saved, although not every justified homicide (e.g. a justified homicide use to prevent a rape in progress, or to shoot an armed criminal who had run out of ammunition even though the private citizen shooter didn't know that fact) would save a life. See also this report on Justified Homicides.
If the ratio of justified homicides to justified shootings not resulting in death are similar for criminals and non-criminals, then the number of justified shootings with firearms is on the order of 900 a year or less (only some of which saved lives).
Private Armed Citizens Rarely Stop Active Shooters
The New York Times has done an analysis restricted to mass shootings summarized in the following image:

This shows that 12 out of 453 active shooter situations (about 2.6%) were resolved by a non-professional firearms owner shooting the active shooter.
Self-Reported Defensive Use Of Firearms Establishes An Upper Bound And Shows The Nature Of Defensive Firearm Use
Survey data from William English measures self-reported defensive firearm use similar to that of Kleck, and with many of the same methodological flaws. But it still provides a data point that places an upper bound on the amount of defensive uses of firearms that occur. It also explores the mix of actions that are seen as defensive uses of firearms in statistical depth.
- William English, "2021 National Firearms Survey: Updated Analysis Including Types of Firearms Owned" SSRN (May 18, 2022) (self-reported data on defensive use of firearms in a survey including "saying you have a gun" as a common defensive use).
English's study shows that most self-reported defensive uses of firearms involve brandishing a firearm, saying you have a firearm without displaying it (31%), or firing a warning shot, but not actually shooting someone.
As noted above, cases of justifiable shooting someone probably happen only about 900 times a year or less in the United States.
Higher Gun Ownership Rates And Fewer Restrictions Increase Homicide Rates
Homicide rates are lower, on average, in places where access to guns is more restricted and firearm ownership rates are lower.
There is no evidence that higher gun ownership rates or reduced restrictions on firearms reduce homicide rates. This statistical trend strongly disfavors the hypothesis that any significant number of homicides are prevented by the self-defensive use of firearms.
Hepburn, Lisa; Hemenway, David. Firearm availability and homicide: A review of the literature. Aggression and Violent Behavior: A Review Journal. 2004; 9:417-40.
Hemenway, David; Miller, Matthew. Firearm availability and homicide rates across 26 high income countries. Journal of Trauma. 2000; 49:985-88.
Miller, Matthew; Azrael, Deborah; Hemenway, David. Household firearm ownership levels and homicide rates across U.S. regions and states, 1988-1997. American Journal of Public Health. 2002; 92:1988-1993.
Miller, Matthew; Azrael, Deborah; Hemenway, David. State-level homicide victimization rates in the U.S. in relation to survey measures of household firearm ownership, 2001-2003. Social Science and Medicine. 2007; 64:656-64.
See also international statistics on homicides and gun ownership compiled by Wikipedia.
Notably, there are no studies showing that self-defensive use of firearms saves many lives, even in places where extremely high levels of public possession and/or carrying of firearms like Israel and Switzerland.
Justified Defensive Use Of Firearms Makes It More Likely That Innocent People Will Be Killed During A Crime
The use of armed self-defense increases the likelihood that an innocent person will be killed in a crime.
For example, insurance companies for banks and other businesses, and corporate managers for these businesses, long ago determined that actuarially, brandishing a firearm makes it more likely that a criminal will kill someone, even if they wouldn't have done so if a firearm was not brandished. See, e.g., here. So, they favor a largely passive response to robberies. Bank employees are trained to comply with a robber's demands. See also 
Despite this passive policy stance towards bank robbery, the percentages of bank robberies that are solved (about 60%) is higher than almost any other kind of crime.
Political Justifications For Firearms That Could Save Lives In Black Swan Events
The U.S. Second Amendment was justified in political terms by a strategy that is currently called in political science lingo "counterbalancing." The leading study on the effectiveness of counterbalancing in achieving the political goals of the Second Amendment with empirical evidence that had not been available to the Founders when the Second Amendment was adopted is the following book:
Lives Lost To Firearm Suicides And Accidents
Finally, because self-defensive uses of guns are so rare, one has to take into account the risks of someone dying from suicide or an accident, with a firearm, and balance that against the lives saved in the rare case where a defensive use of a firearm by a private civilian saves a life.
As the Pew Research Center notes, suicides account for a majority of all firearm deaths, and the number of firearm suicides is fairly tightly correlated with the gun ownership rate. In this regard it is important to note that the literature on suicide clearly establishes that removing one means by which people can commit suicide results in almost no increase in the number of suicides by other means.
So lives are saved every day by simply not making firearms available.
Adult accidental deaths from firearms are quite modest, and in line with all sort of other ordinary activities. But, firearms are the source of a disproportionate share of accidental deaths of children and accidental deaths caused by children, especially younger children. Children with guns almost never use firearms in justifiable self-defense to prevent crimes, however.