That sounds much like an IRC channel with nickname registration services, a talker program similar to EWE-too/NUTS/GAEN, or a simple BBS. Are you wanting a web interface, a specifically client-server interface, a telnet-accessible interface, or something else entirely? You don't mention posts or customizable user rooms, so many BBS, mush, and moo programs sound like overkill.

Something like these BBSes could be stripped down, but I believe all of them are written in C. Citadel and YAWC are open source, and I think you might be able to get DOC code as well. All of the DOC, Citadel, and YAWC BBSes support direct messages to another user while some also allow public chat. Not all BBSes have chat rooms because it was primarily a posting medium. TheBBS I know uses its own code base which is a couple of major rewrites away from YAWC, and it has a nice chat functionality. Telnet to thebbs.nicorinc.com and start an account to see it in action, and you can tell them Mr Mischief recommended the board. I used to have an all-Perl code base similar to it, but that code was lost years ago. It'd be simple to write again if I had the time.

IRC is all about chat, and that's what the 'C' stands for. The 'R' stands for 'relay', because you can have a network of more than one server. There are at least Net::IRC, POE::Component::IRC, POE::Component::Server::IRC, Parse::IRC, and all of their related modules to help with that protocol and command set in Perl. There are also Bot::BasicBot, IRC::Bot, and Infobot among others to handle writing automated IRC bots.

Talkers are often similar to MUDs with no game programmed into them. There are pure chat systems all the way through MUSHes and MOOs that have elaborate room editing, navigation between rooms, and even object editing. http://www.talker.com is a hosting provider that specializes in hosting talkers and MUDs. They have a page that links to the code for many talker and MUD server systems on their site, although some of the links are busted. There used to be good support for talkers on IRC under the alt.talkers groups, such as alt.talkers itself, alt.talkers.programming, and alt.talkers.nuts for example. I doubt there's much traffic there these days, but Google Groups likely has lots of the old info.