If a person sends threatening/abusive/offensive messages to another person via Facebook, Twitter, or any other social networking site, they could be committing an offence. The most relevant offences are 'threatening communications' and 'improper use of a communications network'
For 'threatening communications', the message would have to contain or imply a threat to carry out a serious violent act against a person or persons. This is a relatively serious offence and would not be used for messages that are abusive but do not contain any element of threats.
For the offence of 'improper use of a communications network', the message would have to be grossly offensive, obscene or menacing in character. This is a less serious offence and would cover messages that are abusive but not necessarily threatening.
If you have received any threatening/abusive/offensive messages via a social networking site, and believe that an offence may have been committed, you can report this to your local police force. The police will advise you whether they can progress the matter based on the full facts and your individual situation. However, depending on the circumstances and the nature of the messages, you may wish to initially make a report to Facebook/Twitter etc., as they have processes in place for such situations, and may be able to simply remove the content and/or close down the person's account.
Please see the links in Related Information for Facebook and Twitter's Community Guidelines in relation to dealing with harassment.