New figures show the number of offences involving the social networking sites has soared in recent years.
Last year, Police Scotland compiled 2,710 crime reports that mentioned Facebook or Twitter.
The figure has more than doubled in the last two years, while five years ago there were only 43 crimes linked to the websites.
The statistics, released under freedom of information laws, mean officers are dealing with more than seven complaints involving social media every day.
A breakdown of crime reports showed a wide variety of offences, with the sending of obscene or menacing messages among the most common. There were also numerous sexual offences including grooming, rape, complaints of stalking, allegations of racial abuse and reports of fraud.
Police have investigated alleged online threats aimed at high-profile figures including First Minister Alex Salmond, Harry Potter author JK Rowling and former Celtic manager Neil Lennon.
Politicians said police officers needed to be aware of the dangers caused by cyber criminals. Graeme Pearson MSP, Labour's justice spokesman and a former senior police officer, said: "Given the nature of the internet as an international and unregulated medium, we can expect a growing criminal content in the years ahead.
"Unfortunately predators will seek out the vulnerable, whether it be to defraud them or abuse them sexually, and the internet offers an ideal situation.
"Law enforcement and police will require to beef up their responses to these challenges if we are to protect the vulnerable from exploitation. The consequences otherwise can be severe for victims and their families."
John Lamont MSP, the Scottish Conservatives' chief whip, said: "Presumably the true figure is much higher because police can't possibly detect every online offence.
"People who commit crime on the internet should be treated exactly like those who offend in the real world.
"Just because someone can hide their identity, or cower behind a keyboard, it does not mean they should be shielded from the full consequences of the law."
In total, 6,538 crime reports featuring the two sites have been compiled by Scots police over the last five years. Facebook was mentioned in 5,804 of the complaints and Twitter in 554.
Other offences investigated included attempted extortion, assaults, thefts and breach of the peace.
A Police Scotland spokesman said the force was committed to keeping people safe, and "this very much includes whilst they are online."
He added: "Police Scotland has seen an increase in the involvement of social media and web-based communications in relation to incidents being investigated. This is seen as a reflection of society's adoption of such technologies and its acceptance of the online social networking.
"Police Scotland continues to evolve and develop to meet the needs of the people of Scotland, including online policing and community engagement."
A Twitter spokeswoman said: "Our rules state that you may not use our service for any unlawful purposes or in furtherance of illegal activities.
"International users agree to comply with all local laws regarding online conduct and acceptable content.
"Additionally, our Guidelines for Law Enforcement explain how authorities can request information about Twitter accounts."
No-one from Facebook was available for comment.