Depends on the Library's purpose.
If it is for people to get help editing, critiquing, getting critiqued and improving their stories, then it shouldn't be public.
If it is for people to just post their stories then it can be public if people don't mind using up their "first publication" rights.
I'm not really worried about someone stealing my stories as I am about losing the "first publication" rights. Some publishers are wary of publishing stories that have been posted on a public internet forum, and all the small presses that I know (I do know a few personally) won't consider a story for publication that has been posted on a public forum because the "first publication" rights have already been used.