The Police is only a small part of the wider justice system, and arguably the "easy" one.

BTW, make sure you don't confuse private police with private security.

Private security is employed by YOU, protects YOUR interests and scales in effectiveness with how much you pay them.

Police is employed by a community, protects the community's need to enforce law and order, and it very much shouldn't make a preference for those who pay it more (that would be corruption).

A police force may very well be private, in the sense that it's a privately owned enterprise, but it nevertheless acts out of a mandate of a community and is funded through taxes or a security subscription or whatever it's called.

The hard part is the system of laws and courts. I remain convinced that this benefits from centralisation mainly due to unification. I think laws should be lean but universal.

Imagine an AnCap society where a "community" (defined by a territory with some central governance and jurisdiction) is the size of a city, maybe a county. Imagine going on vacation by car and having to study a dozen law books to make sure you don't break any laws (this already is a problem e.g. in the US where states differ in legality of certain substances and firearms - imagine the laws fragmented even further). Running an international corporation would entail operating in possibly thousands of jurisdictions.