Francis Tandy's Voluntary Socialism, a left-wing market anarchist book from the late 19th century, has an excellent chapter on how law and order would work. (Rothbard's Society Without a State is another must-read.)
“But who will perform the legislative function when the State is abolished?” is another frequent question. “You would surely not entrust that to Pinkerton or his fellows!” Most assuredly not. When a person subscribes to a protective association, a clause might well be inserted in the contract by which the subscriber agrees to serve as a juror whenever he is called upon to do so. When a prisoner is to be tried, a juror will then be selected by lot from among all the subscribers of the association. This jury will then judge the facts, and if they unanimously find the prisoner guilty, they will determine what punishment he shall receive. Strange as such a proposition may seem, it is by no means new. The original jury in ancient times was a means employed by the people to guard themselves against the tyranny of the State. The laws were enacted by the ruling powers, but when they were transgressed, the accused was tried “by his neighbors,” who rendered a verdict, not upon the facts alone, but also upon the law, and decided the penalty in case of conviction. They might find that the facts proved the prisoner guilty of the charges preferred, but that the law was tyrannical, and therefore he was justified in violating it.
A common question is how the environment could be protected in a market anarchist society. There's your answer: civilian juries will likely rule against companies contributing to pollution and environmental problems.