Can people make private servers?
For Quake 3: Arena, yes. The development company, Id Software, released a Dreamcast support expansion for its Quake 3 Server software. It is available to download for free from various hosts on the internet. The servers for most of the online games were run by Sega using their “SEGA SNAP” servers and software. Sega sold the servers and the rights to the software to Nokia, who then renamed the setup “Nokia SNAP”. It’s not impossible that in the future someone may come up with a way to make a private server for some of the other formerly online games, but it is unlikely.