I may be wrong, but I have an impression that ordinary people in Russia (or in Soviet Union, to be more exact) were much less fixated on America as a threat, than Americans on the USSR. The USA was traditionally called "the most possible enemy" by the authorities - a rather phlegmatic and ambivalent phrase, which was not as scary and emotional as "Evil empire". I believe there was mistrust and wariness, but no fear. Something like "Americans don't like us, but they most likely won't try anything, they will not dare".

Moreover, as a kid in the 80's I was much more afraid of Germany (as were many of my friends). America was too distant and almost unreal, and we never were in war with it, while Germany was close, and we knew for a fact that Germans killed millions of our countrymen. Germany was our "Evil Empire", while America seemed like something between a bully and an economic rival (another popular official motto was "(Let's) Catch up with America and leave it behind").