If a person illegally uploads material to a service provider, the service provider is in principle liable for contributory infringement. However, there are those "safe harbors" that reduce the probability of getting legally hammered. The relevant law is in 17 USC 512, which I must point out, is a masterpiece of un-integrated conceptual complexity. Some of the most important themes to be distilled from the law is that to be safe, you have to be unaware of the infringement and you must have specified procedures for taking down infringing material.
IMO the central practical question is, how does this future service differ from existing services like YouTube, and does that difference create greater liability. When it comes time to actually market this product, an DMCA-savvy IP attorney is called for. At the stage of thinking about it, what is probably most important is retaining and exercising the ability to actively and actually responding to takedown notices (see 17 USC 512(c)(1)(C)). If I send you the form telling you that one of your servers infringed my copyright, it better be dealt with quickly ("expeditiously", currently undefined), and you can't say "Sorry, it's out of our control".
As for proxy servers, as I understand it, that is covered by the Transitory Network Communications Safe Harbor, section 512(a), which foregoes the takedown requirements but adds "we had no control" conditions that are generally the case for proxy servers.