Peace movement in USSR was AFAIK only "official" one, promoted by state and party. Participating in 1-May demonstrations, blaming NATO in warmongering, supporting Soviet government in its actions to preserve world peace, stuff like that
. We lived in a small town, our family sometimes participated in such demonstrations - for us it was kind of social activity. To go outside, talk to people, have barbecue in the evening, etc.
People who criticized Soviet government, dissidents, usually did it for the other things. Lack of freedom of speech, press, not enough good quality consumer products, for mistreating Warsaw pact allies, etc. Not for arms race. Generally people felt we were defending side, and almost nobody believed that USSR wanted to start war against NATO. People remembered the previous war which was too devastating.