I figured a branch conversation would be better for our discussion on private law societies. The char limit is larger, and the reply chain doesn't get broken so easy. (Feel free to add people)
The way I see it the final authority would be the arbitration org that was agreed upon in the contract, or perhaps title.
The government is just guns enforcing a decision. A private company could do similar, but many put forward the idea that force would probably be used less than the state currently does to deal with those who decide to ignore a compensation judgement. A tool I've seen mentioned a lot would be an insurance company would go ahead and pay out to the victim parties and then go about collecting from the offending party.