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.
So wouldn't everybody just be better off going to court C in the first place? So court C would naturally emerge as a kind of universal dispenser of justice for everybody, automatically chosen by... guess what... society!
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.
So when the lynchmob comes for me I'll move on to another place and abuse more people. 'cos the lynchmob can't actually lynch me 'cos you're not allowed to be violent. Right?