It's certainly been a long time coming, but the latest version (2.2.1) of Twitter for Mac finally brings support for Notification Center in OS X Mountain Lion. Aside from the side-pane pings for your @replies and DMs, a number of fixes have been implemented that should stabilize your experience some more. Most notably, Growl users on Lion and Mountain Lion can enjoy receiving notifications again. The update is available directly from the Mac App Store right now, so hit the source link if you're interested.
Apple OS X Mountain Lion
Apple OS X Yosemite