Forum plugin with private forum option

Simple:Press

Can limit forums to no access, read only, limited, standard, full, moderator

It’s decently simple, I’ve been using it a bit for my personal blog.
I didn’t have too hard of a time integrating it.
Lots of features available, with pretty fine grained ability to turn things on and off

Also, as information, bbPress no longer requires stand-alone operation
It is now developed as a plugin and is maturing rapidly.
I’ve integrated it into one of my sites with no problems. However, I don’t think it meets your needs. Just wanted to include the info in case you were interested.

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