There are already private courts and private judges, known as arbitrators. Some people dissatisfied with having to wait months for a government court date, instead opt for a binding private court. Works well.
In a free society, this system would replace government courts. Businesses would require that you agree to arbitration from a mutually agreed upon neutral arbitrator (as is the case now). Should you decide not to do so, then they probably won't risk engaging in business with you.
If someone commits a crime against a person or their property, then the contracted protection agency will take such action as required to apprehend you. You're own protection company (and you'd be crazy not to have subscribed to one) will see to it that you are fairly treated and fairly tried. (Obviously, if they don't do this for their customers, they will lose business very quickly.)