Private court system? How does it work? Regular court system works because the government has the threat of violence, has the muscle and rights to hunt, capture and detain. if "private court" rules against me and I refuse to obey, how will they make me obey? There is no "or else" embedded in it so it will be useless.
its a cryptocurrency, the courts legitimacy would likely come from the community forking as necessary to resolve issues as identified by the court, in the same way the us judicial branch gets its legitimacy from the us executive branch employing force as necessary to resolve issues as identified by that court.
its a broader group to convince, sure, but there is a clear 'or else' from which the court can get that legitimacy. 'return the money or else we will return it for you' is a meaningful or else