The Communications Committee said existing criminal laws are "generally appropriate for the prosecution of offences committed using the social media".
"There are aspects of the current statute law which might appropriately be adjusted and certain gaps which might be filled," the Committee said in a report. "We are not however persuaded that it is necessary to create a new set of offences specifically for acts committed using the social media and other information technology."
The Committee also referred to the guidelines drawn by the director of public prosecutions (DPP) in England and Wales for pursuing prosecutions against social media offenders. It said the guidance "appropriately takes account of freedom of expression" rights.
However, it called on the DPP to clarify the circumstances in which an 'indecent communication' "could and should be subject to prosecution" under the Communications Act or Malicious Communications Act.
It also recommended that investigators be given a year, instead of the existing six months, to gather evidence on criminal behaviour on social media before having to decide on whether to bring a prosecution to account for "the frequent need to obtain evidence from abroad".
The Committee said that websites should be forced to ensure that consumers are not allowed to open online accounts without providing identifying information, even if their use of the website thereafter is done so pseudonymously or anonymously.
"There is little point in criminalising certain behaviour and at the same time legitimately making that same behaviour impossible to detect," it said. "We recognise that this is a difficult question, especially as it relates to jurisdiction and enforcement."