Potential war criminal Wernher von Braun, who had been in charge of the Nazi's V2 Rockets - the ones that were fired randomly into civilian populations - worked for the American space program for years after the war because of his expertise. Klaus Barbie, the Gestapo torturer who is estimated to have been responsible for thousands of deaths, was recruited by the U.S. as an agent. I'd imagine that the Soviets would have executed people like these, but it also seems likely that they might have also wanted to use their resourcefulness.
Additional questions: Once they found out, how did the U.S.S.R and its population react to the knowledge that the West was harbouring and using Nazis in this way? And, if the U.S.S.R. was using Nazi's as well, did their public ever find out about it?
Yes, the Soviet space program used captured NAZIs, but the Americans got the top brass and most of the actual equipment including a few V2s, in no small part due to von Braun's actions to ensure that. All the Soviets got "were the rank and file of the V2 program, engineers and minor technicians scattered over the eastern zone." (MccDougall, 45) About 6,000 engineers were shipped off to Siberia where they worked for seven or so years on the Soviet rocket program. Korolev wasn't particularly impressed and didn't feel he had too much to learn from the Germans, he may have felt differently had he got von Braun but we'll never know.
I can't speak to whether or not the average Soviet citizen knew this but considering that the role Korolev played in the Soviet space program was kept pretty quiet I'd doubt they knew about the German 'help'
Source: Walter A. McDougall, The Heavens and the Earth: A Political History of the Space Age