So it depends on who you are talking about. If you mean, the Soviet government — they were spying on the Manhattan Project, and they knew it was going to be used on Japan, but they didn't know exactly when. The response by Stalin was to hole up in his dacha for a day pouting, because he was afraid Japan would surrender before he could declare war on them. When it became clear they weren't doing that immediately, he accelerated his plans to declare war and invade Manchuria. He also immediately accelerated their own atomic bomb program from the relatively small state it was in.

The Soviet press (which was run by the state) did carry a small notice about Hiroshima but carried nothing about Nagasaki. The strategy of the Stalin years was to mostly not talk about nuclear weapons, and if they were brought up, the essential "party line" was a combination of "there is no secret to the atomic bomb" plus "all peace-loving nations should work to control (e.g., ban) nuclear weapons." The former was already a standard line in the West by people arguing in favor of controls, and the latter was fairly disingenuous in that they were developing their own weapons and they were not working towards realistic international control measures in the United Nations (they were basically pushing a plan that was just, "we proclaim the bomb to be banned and promise not to make one," without any inspections or verifications).

Even after the Soviets got their own bomb in 1949, they didn't really talk about it until after Stalin's death. That is another story, but it relates to the question about the general population. It is clear that the general Soviet population knew about the existence of the atomic bomb, but their knowledge of it was very limited as it was not really part of the approved topics for discussion under Stalin. It received very little official public attention; it was not even really condemned in loud terms, because that would just draw attention to the fact that the US had the bomb and (until 1949) the Soviets did not. So the overall effect of this seems to have "muted" the public response until the Khrushchev years. Though, it should be said, it is very hard to measure such things in a society that did not have anything like freedom of expression and where the idea of opinion polling was very much beyond the point.

Two useful books on this subject are Michael Gordin's Red Cloud at Dawn, and Edward Geist's Armageddon Insurance.