Please be aware that when we address questions of this nature, we cannot speak to hypothetical situations, nor can we guess what we would have done in a situation where we were not involved. Nor we can offer you legal advice.
In the United States, the Digital Millennium Copyright Act (DMCA) defines the process by which a copyright holder can request that material be removed. In such cases, a provider is legally obligated to remove the allegedly infringing material without judging the merits of the claim. It is essentially done on the copyright holder's sworn say-so.
However, the DMCA also governs the process for restoring material, and that process is similarly rubber-stamp on the part of the service provider. Once that's done, it becomes a matter for the courts, not the service provider.
In this regard, the DMCA is a good law for us and for you. (Although it can and does suck in many areas, this isn't one of them.) The DMCA is, in part, supposed to protect you from capricious decisions made by the service provider based on some subset of the facts. At no point does or should an Internet service provider investigate or make judgment calls about complex copyright law and questions of what might be infringing. (We specialize in server processes, not process servers.) The DMCA gives us (and you) a certain (non-perfect) confidence that the copyright owner's claim has at least some legitimacy, and provides decent protection (once you get over the initial take-down hurdle) against the use of false claims of copyright infringement to suppress legitimate content.
If you wish to host a controversial site in the US, it behooves you to know the law, particularly this one, and how to use it to your advantage in the event of a dispute. You should also be prepared for a downtime of some or all "allegedly infringing" material for a couple of weeks if the copyright owner wants to fight.
We adhere to the entire law very closely. We do not generally pull the plug on an entire site if, for example, someone claims that a single graphic is infringing. We do our best to remove only the content that the copyright owner specifically identifies as allegedly infringing -- usually by allowing you to handle it yourself unless you decline or fail to do so. We allow and encourage the use of the "putback notification" process when material is incorrectly identified as infringing. But we do not automatically terminate a member's service merely for receiving a complaint alleging infringement. (However, actually infringing someone's copyright does violate our TACOS and will generally result in immediate termination.)
Keep in mind that while we aren't lawyers, neither are we idiots. We can tell the difference between people harassing our members via the DMCA and cases where our service is genuinely being misused, and we can adjust our attitude accordingly. Fortunately, both of these cases are very rare.