You pay your taxes, don't you? Why wouldn't you pay private firms instead to do the same job more effectively? And then, of course, it's easy to imagine that such a regulator would only provide information to paying customers using a smartphone app, for example.
> How do we know the regulators aren't influenced by whichever of the protection firms they buy from?
How do you know your government regulator is not in bed with the company it regulates? In fact, you don't and in fact most of them are. You can only elect government officials (and not regulating agencies themselves) and then they stay in power for n years almost indefinitely.
On the free market, there's competition of regulators. If I suspect one regulator takes bribes, I switch to another. Or maybe I subscribe to both and compare their info. Or alternatively, if I saw they were all in bed with consumer businesses, I'd stop paying them completely, as would many people, and that means they'd go out of business.
> If people are apathetic about the property rights of their neighbour what right does the government have to step in and stop them taking that property?
This one I didn't understand. In fact your last paragraph looks very odd and I don't understand how you came to imagine such a scenario. One thing I can tell you is that the poor will be able to afford protection agencies, just like they can afford owning a car or a pair of jeans or a computer. A protection agency doesn't have to be expensive to be profitable and it doesn't mean having personal body guards. There are various segments on that market and each can be covered. The crucial difference is that the poor will now be customers of the protection agency, whereas in the current system, I guess you're not going to argue, it is the poor who most often become victims of police brutality and injustice - the reason is, they're not customers, the police and the judges will be paid indefinitely and they know it.