I think what you're mostly looking for is an Instant Messenger script, but rather than a client, it's used by the site...much like Facebook.
It'd be really difficult to make a plugin like that without actually making it a software. You MIGHT be able to download AJAX Chat, and install it to MyBB, then set it up so they (the users) can can immediately enter their private channels, and invite users to chat from there. But AJAX Chat is also bloatware imo.
Well, did a bit of digging myself. Found this pretty neat IM Script. Intra Messenger.
http://www.theuds.com/intramessenger.php?lang=EN
The admin control panel for it
Demo:
http://www.theuds.com/intramessenger/adm.../?lang=EN&
Was incredibly functional, and had many many useful systems. For instance, the ability for the admin to display a message to all users (whether they be offline, or online, or either or) whenever they wished. It's a really beautiful piece of work, and even better, they had tutorials on how to make it use the user databases of forums (including MyBB), CMS's (like Joomla/Word Press), etc. Seems very nice.
Only problem is, it uses an off-site client, which means users wouldn't be using it from your site, but rather the client which they would have to download. But the nice thing is the developers released it as FREE for personal, community, and commercial uses all. (which also means you can host the client download on your site). It also appears to have no limitations/restrictions on use (so long as credit is given I think)
Screenshots:
http://www.theuds.com/intramessenger.php...ts&lang=EN
If you try to use it though, I recommend backing up databases of course. Always gotta do that.
If that doesn't interest you, dunno what to tell you. Would be both tedious and slightly difficult to make a plugin for a Facebook style chat. Nickman released a nice IM type plugin, but it has various quirks, and not the advancements of a chat like Facebook.