i think the legal issues surrounding private servers are similar to console emulation. As long as you write your own bios/dont provide a bios your not breaking any laws, same way if you write your own server code for a private server. Also as a private server makes use of a commercial game client, the private server owner must also not provide pirated copies of the client or charge money to use their server.
An example,dating back to 1999 a commercial Playstation emulator was released called Bleem they wrote their own bios and it allowed you to play ps games on your pc. Obviously Sony sued and Sony actually lost! unfortunately Bleem was bleeded dry by legal fees and ended up going bust anyway! So that's what happens with private servers, they get sent cease and desist letters and because they don't have the money to fight they shut down... or maybe they were actually using leaked server code rather than their own which is definitely illegal. ironically one of the guys who worked on bleem actually now works for sony on backwards compatibility for games on playstation store!