Answer:
Government is predicated upon the initiation of force. For example, if you were to stop paying your taxes, you would get a letter from the IRS saying you owe such and such amount. They would keep sending you these letters for a while, but eventually they would send a group of men with badges and guns to throw you in a cage for not giving this institution YOUR money.
Courts are often crowded because of overcriminalization (prosecuting individuals for victimless crimes) and there is no price associated with criminal cases. Cases often get plea bargained because prosecutors have an incentive to have high conviction rates, not to serve their clients since they are paid by the state. This often means that crimes that are easier to prosecute get more attention.
Also you have the current problem where people sue each other willy nilly. This problem could be solved by simply requiring the person who brings suit to pay for the defendant's legal expenses if the defendant is proven innocent of any wrongdoing.
Also the victim of a crime in our current system gets screwed twice. Once by the offender who perpetrated the crime, then again by the state, which confiscates his wealth in order to pay for the prosecution and housing of the criminal. Historically private law was centered around compensating the victim financially.
In absence of a state professional judges would emerge that would make decisions based on a commonly established practices which could be codified into a law book. This would be analogous to how dictionaries codify definitions for words, you don’t need a central body to come up with new words and definitions, language emerges spontaneously. So too, law would be established in this way and in fact this is what English common law is based on, communal law that was established and accepted by the community. Similarly customs would emerge for the amount of times you can appeal, based on the crime committed.
Private law would solve many of the above problems listed. It would eliminate overcriminalization (it would be unlikely that someone would bring suit against someone for using drugs), it would focus on restitution and lawyers would be working for the victims instead of simply working for a high conviction rate.
