Private citizens defending with guns turn out to be much less likely to accidentally shoot the wrong person than the police are. The private defender has a number of key advantages over the police, including these:
(1) Response time. The private local defender is present at the scene (or near the scene) when crime happens, whereas the police have to be called in and come there from elsewhere. It is quite likely the private defender actually saw the crime or criminal attempt, whereas the policeman has outdated third-hand information conveyed through a few intermediaries over an unreliable phone and/or radio connection.
(2) Geographic knowledge. As somebody who lives or works near the scene of the crime, private defenders are intimately familiar with the territory. They know where the entrances and exits are; it's hard to catch them by surprise. They have "home field advantage", to use a sports metaphor, whereas the police arriving in a hurry after the fact do not.
(3) Human knowledge. As a local, the private defender knows the other locals. The boss, the employees, the loyal customers, the shopkeeper next door - the defender knows all those faces! Policemen arriving in a hurry after the fact do not; they have to make a snap decision who the bad guys are.
(4) Flexibility. The private defender doesn't have to attack - it's not his job to do so. So if he's pinned down or exposed or if attacking seems impractical for any reason at all, he has the option of simply not defending. Letting the criminal succeed and leave is still an option. Waiting for the cops to arrive and letting them handle it is also still an option.
(5) Liability incentives. A private defender can be sued into bankruptcy if they shoot someone by mistake - cops can't. The private defender can be held responsible for any property damage they do; cops can't.
In summary; if individual private citizens have the freedom to defend against criminals, they tend to use that freedom pretty productively. They can stop crimes by virtue of being present and well-informed and well-prepared and properly-incentivized to produce a remarkably low chance of causing collateral damage. Cops show up late, are ill-informed about the original crime, don't know the territory, don't know the people, aren't held responsible for their actions, so surprisingly often they end up shooting into the crowd, needlessly killing dogs, needlessly damaging property, and just generally causing more violence and mayhem than was necessary to address the situation.