Most free market proponents are minarchists, and I think they're sending a flawed message out there - that we ought to rely on die-while-you-wait government courts to resolve disputes that occur (i.e. in the absence of regulation). Where I live, there is a government program that provides "comprehensive, no-fault personal injury cover". It was established precisely because the government courts are so terrible (another example of snowballing socialism).
Private courts could work (better) in a minarchist society:
Most (if not all) business contracts would stipulate a third party arbitrator in the event of a dispute. Consumers and businesses would need to have liability insurance, in case they decided not to pay damages ordered by a private arbitrator. In order to get liability insurance, you would need to have a good liability rating.
In a free market society, there would be a greater sense of community, because neighborhoods would have to negotiate the shared use and maintenance of roads, pipes and cables (perhaps gated communities would become normal in this society). Those neighborhoods that did not also have a set agreement on private arbitrators in the event of a property dispute, would be at a competitive disadvantage in the housing market.
Likewise, if a partner did not agree on a private arbitrator in the event of a family dispute, they would be less desirable as a partner. Buyer beware.
What about pollution?
The use of air and water can be negotiated by private protection agencies, and treated like private property. However, it would (at first) seem that a minarchist society could not tolerate the privatization of environmental civil law, because harmful pollution is like a stranger throwing a brick through your window. You probably never agreed to a third party arbitrator with such a person. But that's not the full story, and I'll dwell on it later.