How a private law society would work or has worked in the past has been laid out by the people I recommended you to read: Hoppe and Rothbard.

I referred them to you because their literature will be able to lay it out to you in a much better and systematic way than I can in a Nostr thread with the limited time I have available to me for that purpose.

I think they can add a lot of value to and enrich your reasoning while thinking about governments and law. I hope you find the time to read them someday.

'For a New Liberty' by Murray Rothbard is a great book to start. There is a chapter about private law in it as well.