what are private law proceedings?

Answer 1

No what the poster above me is describing, is "arbitrage". (Not sure if it's that word in english but I think so).
Private law proceedings are law proceedings concerning private law. And private law stands opposod to public law. Public law basically regulates the relationships between citizens and the authorities while private law regulates the relationships between citizens.
So private law proceedings are the kind of proceedings that occur between two private parties. Actions for damages, or divorce, or...

Answer 2

not sure if this is what you mean, but a growing number of litigants are settling their cases through private mediation--if A sues B, the parties agree to hire a mediator, often a retired judge, who meets with the parties, studies the case and issues a ruling. If one of the parties doesn't like the ruling they can wait until their cases comes up in the regular court system, but the "private judge" system is popular with big companies especially because it's a lot faster than the regular courts.