First, consider a criminal case.
Suppose person A hires police/court XYZ and person B hires police/court UVW.
If XYZ and UVW agree on the proper resolution, then there’s no problem.
If XYZ and UVW disagree, then they hire a third court, RST. Then, RST resolves the dispute.
In a free market, police protection and insurance may be bundled together. If necessary, XYZ will compensate A for their loss while UVW compensates B for his loss, if it’s truly impossible to determine who is responsible.
Also remember that “compensation-based justice” is different from “revenge-based justice”. For example, in the present, a court can only rule “guilty” or “not guilty”. In a free market court, there could be a ruling that says “With probability 10%, B murdered A”, in which case B owes 10% of the fine/penalty for a murder. In that case, XYZ would be responsible for reimbursing A for the remaining 90%.
XYZ and UVW won’t let a dispute escalate to all-out war. If they did that, they would lose their customers. If the dispute was truly un-resolvable, they would write it off as a loss. If UVW made biased rulings in favor of its customers, then it would be shut down by its competitors. In the present, the State is a corrupt protection agency that makes biased rulings in favor of its customers. If you say “What happens if a rogue police agency imposes its will violently?” The answer is that, if that happens, you have a State, which is exactly what’s happening now.
Second, consider a civil case, regarding breach of contract. In that case, a contract would usually specify what court had jurisdiction.
Free market competition ensures fairness of courts. All decisions are public and scrutinized. If a court habitually makes bad decisions, it will lose its customers.
In the present, judges and State employees are absolutely immune if they do something wrong. As the above example stated, suppose a person is wrongly convicted of a crime and spends 10 years in jail. The police, prosecutor, and judge face no negative consequences for their mistake. The victim usually can’t collect compensation for the 10 stolen years of his life. Suppose Congress passes an illegitimate law banning possession of marijuana. The people who write and enforce illegitimate laws suffer no negative consequences.