And unmoderated share links should probably be noindex and nofollow anyway, no? Why would you want search engines to crawl that?
Personally I would expect share links to be unlisted (not indexed by search engines, not publicly discoverable, with a long enough uuid to be effectively unguessable). "Publish publicly" could be a separate function you add later, with moderation?
You're right though that you have to moderate the stuff. It's typical backwards Google Economy thinking that have to block spam to please Google (that's how an SEO spammer thinks, it's the only thing they think) as opposed to block spam so that your users will keep coming back. Letting your web site get spammed is like fertilizing your flowers with herbicide.
But yeah, I think spam should be deleted altogether, not merely hidden from search engines.