Try fiddling with the Desktop Audio Boost setting. I would suggest you do this on a local recording only so you don't blow up any viewer's eardrums. It's going to be a pain in the ass to get right like any major audio changes, of course.
Alternatively, Mumble has a neat feature where it lowers the volume of everything else by 80% when someone is talking. I'm not sure if Teamspeak has such a thing, or if you'd be willing to change to a Mumble server, but it sounds like it may be a solution to your problem, too.
