Has privately funded national policing ever been implemented?
In a low-tax or even zero tax system, one would assume there would still need to be some kind of legal system in place and the only way I can think of to enforce that isn't simply mob rule is a national, privately funded police system.
Has this ever been tried and if so what were the results of it?
I can't think of any case where this has been done at a national level (but see one example below from the Middle Ages), although some similar circumstances exist at the local level.
For example, in Colorado, there is a municipality called "Lakeside", in metropolitan Denver, which is home to an amusement park, a strip mall, a little lake, and a couple of small apartment buildings on perhaps one or two square miles of land.
All of the land in Lakeside is owned by one person and all of the taxes imposed by Lakeside are paid by that one person who controls all of the elected offices, effectively sets the amount of taxes that will be imposed, and finances all municipal government operations including a police force which maintains law and order with full governmental authority on the property owner's property. (The current owner is the elderly daughter of the original sole owner of all of the property in this municipality, so she actually received her power on a hereditary basis.)
Nominally, this isn't privately funded, but in substance, it is privately funded by the sole property owner in the municipality.
Historically, Pakistan had similar arrangements on a much larger geographic scale in its Northwest Frontier provinces, although as I understand it this is no longer the case, and English gentry similar had law enforcement authority ex officio (and at their own expense) in the vicinity of their estates through the 19th century.
Certainly, many law enforcement agencies at all levels receive some private grant funding for particular programs.
There is a city in India that is owned entirely by one company and resorts almost entirely to its own private security force in lieu of governmental police, and there is a religiously affiliated compound/village in Africa that operates on a similar basis. But, again, this is not true at the national level in any country of which I am aware.
Also, while privately run or funded police forces at the national level are unknown, for profit prisons are fairly common in the United States and these are often partially financed through inmate fines and fees rather than appropriations from tax dollars, although they are almost never entirely privately funded.
Certain kinds of regulators, such as tax collectors, have historically been financed, in whole or in part, from a share of the revenues they collect in the course of their duties (something that persists today with civil forfeiture statutes that benefit the collecting agencies and municipal law enforcement agencies funded primarily with fines they collect), but this really isn't private financing either.
Some governments, at all levels, have volunteer auxiliary law enforcement officials, which were historically known as "night watches" or "militias" or "posses" which received minimal or no government funding and instead financed themselves, although today, volunteers primarily act as reserve forces for overflow needs, rather than as the primary source of law enforcement.
UPDATE: The one well known case of privately funded national policing that existed historically, although it does not now, was Iceland in the Middle Ages. Then, governmental courts would authorize privately funded law enforcement actions against individuals based upon privately developed and funded evidence presented to them by a victim, and the privately funded law enforcers would carry out its will, and this was national in scope (although Iceland, even today, has a population of only 100,000 and not a huge amount of territory (40,000 square miles) and was even less populous ca. 1000 CE when this regime called the Icelandic Commonwealth (population 50,000) was in place). This small population was divided into about a dozen clan-like Chiefdomships who organized blood feuds and managed cases in their court system.
After about 300 years this system failed to function and the King of Norway came in and imposed order and ruled Iceland for many centuries thereafter.
