I’m not aware of any way so that You may post public content (be it anything) on a Private Forum. a Private forum is usually meant for internal discussions etc for large institutions. Usually with a 3rd party SSO etc (though it works just fine without that too).
When We’re talking about public content, That is Public and hence, Category can’t discriminate it on a per-post basis.
I hope someone from development team may have a hack better than I suggested but AFAIK, I’m not aware of any such way.