People communicate in different ways and for many, communicating via email has had its set of issues. Over the years, people started to use email as a way to harass others. It got so bad that many states passed laws that made it a misdemeanor to send e-mail "with the intent to harass". Normally, the line gets crossed when multiple emails (or tweets) are involved that are intentionally abusive.
Sometimes, a twitter chat can turn unexpectedly ugly. If you get harassed on Twitter and the person persists in their abusive behavior, the best thing to do is not to respond. If it is an associate you work with, you can report it to your HR department. If it is someone you don't work with, you can block the user and report them to Twitter via this link. If it involves real threats, call your local police department.
It is clear that the day is coming when states will update their legislation to make harassment via Twitter the same as email.