Invite-only vs open & public?

In order to minimize day-to-day management of the wild Internet, I have it set to invite-only at the moment. It does cause friction with people joining, which has its own trade-offs. We may want to relax this requirement.

Secondly, how much of the forum should we have behind logins? I am privacy-minded, though I could see an argument for it being public and searchable by Google, et al.