Hendon Labour MP Andrew Dismore raised the matter in a special Commons debate, making a personal attack on a member of Conservative-run Barnet LBC.
The MP singled out deputy leader Brian Coleman, who is also a Greater London Authority member.
'I am concerned he is using correspondence from the opposition,' Mr Dismore told the Commons.
'I believe he is behind attempts to create a Dame Shirley Porter-style regime, which was branded at the time by the district auditor in Westminster as creating a climate of fear.'
Mr Dismore, who spent 15 years as a member of Westminster LBC, said when his correspondence was routinely copied to councillors and officers it was a breach of confidentiality.