How does the availability of an illegal copy of copyrighted material affect legality and liability?

The users of the website I am designing will be storing chunks of written content on their profile for personal/collaborative reference. This content could potentially be copyrighted ("CR"). What level of caution do I need to have? As I understand things, it is illegal to make a copy of CR content, except for "fair use" (assuming no CC license or the like). Is private use "fair use"? If it were impossible for the user to share the content (other than by giving out his account credentials), do I even need to worry about liability, since it would be impossible for copyright holders to locate that content? Or could they still come after me if they thought my users were storing CR content? If I implement networking, does that drastically change things? Non-friends would have no access, but a user could choose to share content with friends. So CR content wouldn't by default be redistributed, and a copyright holder couldn't see this content unless he were friended and sent a share link.