A: It depends! This is a minefield, so tread very carefully, and preferably with guidance from a good internet and copyright lawyer. Here are some of the laws and issues to consider:
If the service actively encourages users to upload copyrighted music and videos, then it could be found liable for inducing or contributing to the users' copyright infringement. This is what killed Napster and other file-sharing services. If there's no such encouragement, then that is less likely—but still don't rule it out. It's very fact-specific.
The Digital Millennium Copyright Act provides a safe harbor for online service provides, but only if they comply with the notice-and-takedown framework it provides. This is why YouTube stays alive: when copyright owners notify them of infringing videos, they take them down right away. It's very challenging to do this at scale.
Beyond copyright issues, Section 230 of the Communications Decency Act provides immunity to online service providers. This would protect against claims like defamation in content uploaded by users.
Whether the sharing is public or private makes little difference, except perhaps for the number of copies and therefore the magnitude of potential infringement.
A: A well-written and specific cease-and-desist letter is usually a good first step. It will be taken far more seriously if it comes from a lawyer. Keep in mind that trademarks can be lost through lack of policing—so sending these letters is important to that end as well.
A: There is some new information on this question as of July 2014. Judge Jed S. Rakoff in the Southern District of New York issued a more detailed opinion which found that collecting legal briefs by lawyers (not just opinions and orders by judges) was fair use.
Professor Eric Goldman summarizes the issues and the case, which is White v. Westlaw: