Who chooses which private court? Presumably there will be many, each catering to different markets for justice, and even some competing in the same market.
Exactly right. Let's say that we have a disagreement and we each go to our courts, my court A and your court B. There are four possible outcomes.
1. A and B both rule in my favor
2. A and B both rule in your favor
3. A rules in your favor and B rules in my favor
4. A rules in my favor and B rules in you favor
Outcomes (1) and (2) present no problem. The matter is settled. Outcome (3) is where each of our courts rule against us. That's unlikely so we'll ignore that. The real issue I'm betting you're worried about is (4) where my court rules in my favor and your court rules in your favor. There are then two additional possibilities.
1. A and B have a mechanism to settle this by going to court C to settle the dispute
2. A and B don't have a mechanism in place and simply go to war with each other
We'll call the courts that have a mechanism in place "legitimate courts" and we'll call the courts that don't have a mechanism in place "bandit courts". Since as you pointed out, there will be competition, who has the advantage? I'll quote Walter Block on this.
For suppose there are two types of courts: those who will mediate, when they find themselves on opposite sides of a decision from another court (call them the "legitimate" courts), and those who will not (call them the "bandit" courts). The latter, it is clear, will have to fight in every case. The former, only when faced with one of the latter. The point is, there will be a competitive advantage enjoyed by the legitimate courts vis a vis the bandits. Fighting is expensive. A firm that regularly engages in such activities will suffer additional costs for ammunition, tanks, planes, to say nothing of combat pay for its employees. Bandit courts will have to fight all of the time, legitimate ones only some of the time. There will be a tendency, therefore, for the legitimate firms to out compete the bandits. This furnishes yet an additional reason for expecting bandits to be very much the exception to the rule. For if even would be or potential bandit courts realize they are likely to be consigned to the dust bin of economics, they will be less likely to start on this path than otherwise.
But who can stop me opening a restaurant called "Burger King", with the same sign over the door, the same color seats, the same menu, the same everything - except (maybe) the food?
If you defraud me into thinking that your restaurant is some other restaurant or that I'm buying a CD from Trent Reznor when I'm really buying it from Rent Treznor, I'll sue you in the aforementioned private courts. If you're not committing fraud, if I'm just a consumer not doing his homework, well,
caveat emptor.