I ask because there were protests in the west against nuclear armourment.
However in the U.S.S.R. they did not have freedom of the press and despite them testing weapons and having nuclear power plants I see no reason for them to tell the people they could destroy humanity - it might even be quite destablising. So how would they find out?
Under Stalin, the Soviet Union said very little publicly about its nuclear capability. They kept things deliberately confused, having announced in 1947 that there was no secret of the atomic bomb, but not announcing that in 1949 that they had finally tested one. The fact that Western sources — including over radio waves that Soviet citizen could access — reported the Soviet test however would have made it pretty clear to your average Soviet citizen that they had nukes and could in some way threaten the United States and/or its allies if they were themselves attacked. But the number, kind, etc., would not have been discussed. Only after Stalin's death did the Soviet Union have a somewhat more open policy in talking its nuclear weapons (open relative to what came before, anyway).
As for why they might have or might not have talked to them about it — the Soviet people knew, of course, that the United States had nuclear weapons. So the truly destabilizing thing would have been for them not to have some way to deter American aggression as they imagined it. What is interesting about Stalin is that he really did not want the threat of nuclear weapons to be taken all that seriously by the people, which is why they made basically no preparations for any kind of attack until after his death. He knew that even with the bomb, the Soviet Union was very vulnerable to an American attack, and sought to just avoid acknowledging that. After his death, Khrushchev's response was to be more open about these things, but at the same time also exaggerate Soviet capabilities in the other direction, emphasizing a parity and capability that they did not yet have (the Soviets would not really achieve a survivable, credible capability to hit the United States until the 1960s, and would not achieve something like nuclear parity until the 1970s).
The best source on the Soviet nuclear program in these years is David Holloway, Stalin and the Bomb, but Ed Geist's Armageddon Insurance also talks a lot about what was and wasn't said about nuclear weapons in the USSR.