Mac OS X : Mountain Lion 10.8 supports twitter account mangement and sharing via default application on the OS with other mail accounts and iCloud feature they already supported in previous versions.
If you haven't set up your Twitter account yet - you'll want to go into system preferences > Choose Mail, contacts & calendars to set up your various accounts.
Loggin into your twitter account is just easy as loggin into the main site, and especially if you share information and links often on social networks, it could make things a lot easier.
Apple didn't included the facebook support yet on their official release of OS X 10.8 yet. It was available during the beta version of the OS but due to some reason Apple team decided to get rid of it for now. Facebook support will not be available untill the next software update in this fall. Also, as has been pointed out on Twitter itself, the Mountain Lion tweeting option via the Share button opens the Twitter website only -- it doesn't (yet) work with native clients like TweetDeck, Twitter for Mac or YoruFukurou.
Disappointing Mt. Lion detail: clicking Twitter notification banner takes you to Twitter.com website, even if app is installed.— John Gruber (@gruber) July 25, 2012