I used to moderate a small internet forum. It was relatively peaceful, but we got the odd reported message from time to time, reported by our forum users, when there were suspected spambots, arguments devolving into personal insults and one person in particular who used to get hysterical about the tiniest of things, hurl abuse at anyone she thought deserved it, got banned and then came back with a new account to be banned again. Banning her took all of five minutes out of my day when it happened.
Twitter can take care of this, it's just that they won't. Even 4chan, pretty much the most awful den of iniquity on the internet, is capable of banning people and their posters are supposed to be anonymous. It's never anonymous, someone out there with the necessary skill can track you down if they feel the need. See here for details:
www.cracked.com/article_17170_8-awesome-cases-internet-vigilantism.html
What twitter should do, in good faith, is the same thing facebook does on a wider scale: hire people trained to crack down on trolling and get them to do it, day in and day out. Hire lots of them. Problem is, that's a good moral decision but not a good business decision. It would cost them millions in training and wages. Even outside of the law, it's not hard for a social media giant to have its rules and enforce them. Mumsnet does it, Tumblr does it, Pinterest does it brilliantly, Facebook not so well but they try, and 4chan does it best because those people can be so bloody-minded and have some of the best hackers on the internet.
A public naming and shaming would do the job nicely. The men with families and a good public image would have to put up with being the shame of their families and the ones who don't care would be weeded out as criminals and prosecuted once they go too far.