If you're like me and use another Twitter client on your Mac running Mountain Lion, you might be annoyed with the double notifications you receive regarding mentions and direct messages. Moreover, you might be annoyed with the fact Mountain Lion wants to use Twitter's website to reply to tweets and DMs. What to do?
Luckily, there is way to disable those notifications. In System Preferences, click Notifications. In the list on the left, find the Twitter line and uncheck the three boxes in the listing on the right. In addition to this (and this is very important), click the Options… button under the checkboxes. An overlay window will slide down asking if you want to disable notifications for mentions and direct messages. Uncheck both boxes. This will stop Mountain Lion from generating double notifications when you get some kind of incoming message on your account.
Currently, I'm testing out the TweetBot alpha for OS X and I'm impressed. However, the double notifications generated by Mountain Lion were becoming increasingly annoying. They were getting to the point where it would downright bury the notifications from TweetBot in the list in favor of the official Twitter notifications. Not cool. After a little digging, I found there was an extra step I needed to take (the Options button) and voila - problem solved!