I wouldn’t even really call this a private forum, though. Anybody can join for free with minimal effort.

Now, SomethingAwful is a private forum, I paid them ten bucks to join. And to be honest I like their style. It’s a cheap one time payment for permanent membership but if you’re being an asshole and get yourself perma banned, you’re going to pay another tenner to rejoin. It’s a great anti-troll and anti-botting feature.

I don’t think that strategy would work well for the Fediverse and I don’t recommend it as such. But that’s what I’d consider a “private forum”.