You kind of answered your own question right there. You’ve already made a decision that at least some portion of the site will be publicly viewable, now it is just a matter of how much. Your circumstances are a little unique but I think I would want at least some portion to be open. A landing area, basic info and FAQ. Some folks are nervous about joining a forum or facebook group, (I’m sure you’ve heard the forum term ‘lurker’ - folks who read the forum everyday but never join or post) do you not want to help those folks? of course not we all love cats!
So a possible solution might be to make some type of visual difference between the public and private portions of the site. Something to warn a user that they have left the private portion of the site and are now posting in a topic which is public. This idea taken to it’s extreme would be to style the site to actually appear to be two completely different websites.
It wouldn’t be hard to do because all the private portions of the site require logging in. Once a user logs in, they can be re-direct to a different homepage which could be styled in a completely different way then the public portions of the site