Direct Messaging On Twitter
A tweet isn't anything to be worried about—it's as innocent as can be. Being angry that your girl is tweeting another guy is ridiculous and will essentially make you look like a jealous jerk. Everyone and their mother can see what she's saying and to whom she's saying it. Obviously, if she's tweeting at some dude and telling him that he has a great dong, sure, proceed to flip the hell out. But unless your girlfriend is dumb as dirt, that's an unlikely scenario.
However, direct messaging on Twitter is totally different. Direct messaging someone on Twitter means that you want to say something to the person, but you don't want anyone else seeing it. By definition, it's a secret conversation, one that can absolutely be shady. If you discover that your significant other has been DM-ing a new dude quite a bit, brace yourself for the worst.