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No details were given of the incidents which have reportedly involved local SNP councillors.
The investigation was revealed as it emerged new Midlothian councillors are to be given extra advice on personal safety in the wake of the death of MP David Amess.
Councillors unanimously backed a motion calling for elected members to receive the same support and security as national politicians.
Councillor Joe Wallace, who brought the motion, said: “We as councillors are just as vulnerable as MPs and MSPs.
“We are easily recognised in the streets because we gladly put our photographs through people’s doors when it comes to election time, we can be reported in the media and words can be, and have been, twisted and taken out of context for political gain, and for modern methods of communication can be targeted at a particular audience or group.
“Unfortunately we have radicalised, hateful, bigoted people amongst us and when the rich and the powerful are seen to act with impunity they think they can do likewise”
Mr Wallace said he and his councillor colleague Debbi McCall, along with MSP Christine Grahame, had all been targeted by “verbal abuse, harassment and attempts at intimidation”.
He said: “I cannot say too much about it as it is in the hands of the procurator fiscal, however I have particular reason to thank Police Scotland and the fire rescue for providing me with particular advice on personal safety.”
Councillor Janet Lay Douglas told of her experience not long after being elected in 2017.
She said: “As a new councillor I experienced in the early days a stalking incident which I left too long.
“It was about six months before I raised the issue with the council.
“I do feel if there are any new councillors coming in it would be beneficial to have something there on how to protect themselves”
Council officers agreed to include additional personal safety advice in new member packs which will be handed out following the council elections next year.