People ask this periodically on here, and it's an odd question since the entire Stalinist period in the Soviet Union was characterized by a "terror" that was directed at multiple "rightist" groups within the country, including Mensheviks, Trotskyites, kulaks, engineers (!), and a multitude of other miscellaneous groups who were accused of waging "anti-Soviet" acts of sabotage against the nation. Unlike in the American Red Scare, these people didn't just lose their jobs and end up on black lists (which is awful and undemocratic, I agree), they often got sent to labor camps or even just summarily executed.

I think the disconnect is that people imagine that the "Soviet version of the Red Scare" would be Soviets imagining that US agents were undermining their society, or that cultural output was being subtly influenced by pro-democracy forces, but this totally misunderstands the asymmetry of the situation. The fears weren't about foreign agents infiltrating the Soviet Union — in a totalitarian nation with a hard border and massive bureaucracy (to say nothing of agents abroad) this just wasn't a problem. The few times the US tried to send in outsiders, the outsiders just got captured and shot, immediately.

The fears of the USSR were of internal subversion, usually not from people thought to advocate an overthrow of the Communist system — that was so obviously subversive that only someone with a death wish was going to do that during the Stalin period — but from the belief that alternative ideologies (many of them, like Trotskyism, were still Communist!) that deviated in core ways from accepted Stalinist dogma would undermine the Soviet "project."

Fears of foreign influence was of course an issue, and played a major part in some of the ideological battles fought (notably in the Soviet academic system, where foreign scientists were at times cast as "bourgeois" influences whose work should be viewed suspiciously), so that's somewhat of a parallel with the US Red Scare. But the Soviet version was a much more dangerous game, and went on for a much longer time — decades of "terror," millions dead or imprisoned, etc.