There are two possibilities here: first, both sides want this solved. Second, only one side is willing to solve the problem (i.e. one side thinks he dindu nuffin)
In the first case, they'll choose a judge they both agree to be impartial. Before anything starts, they agree to accept the results. Not accepting would be a breach of contract, which would be another conflict to be solved. In theory, this could go on forever, with one side never agreeing with the results. Which would lead to something equivalent to the second scenario.
The second scenario, only one party goes to a judge. He will seek the most respected and impartial judge, as to assert the most legitimacy to his claims. After that, he's in charge of seeking reparations or indemnities from the other side.
Assuming citizen B really committed a crime against citizen A, then citizen A only needs to prove citizen B's crimes against the laws of his people. The judges, respected by the people, are only there to confirm citizen A's grievances are legit.
If there are two judges, both respected by the people, but they disagree on whether or not citizen A's grievances are valid, they will compete with eachother for the best case, the one that convinces the most people within the social circles they are relevant.
In the end, it boils down to legitimacy, and what people consider it to be. Some people consider it to be legitimate to make a living scamming old ladies with expired lottery tickets, others will consider this to be fraud, a crime.
With no monopoly on justice, conflicts are dealt locally, according to local laws. With a government monopoly on justice, conflicts become much larger than what they were supposed to be, because you have people from city A writing legislation for the people of city B, without ever knowing what laws the people of city B consider to be fair and just.