They don't. They can't without buying themselves a whole lot more trouble.

LJ is in the same category with ISPs and on-line storage sites like YouTube or MediaFire. They are not responsible for what someone has posted/uploaded until they are made aware of some kind of violation - usually through a complaint. This protects them from being sued by Scholastic and JK Rowling among many others.

If they actually trolled the site looking for violations they could be reclassified as "online publishers," in which case they would be responsible for *everything* that *anyone* posted on the site.