Yes, definitely. The Soviet citizens knowledge of the space race was not nearly as open as in America for example, due to the Soviet propaganda machine having firm and total control over the press.

Still though, the Soviets achieved many notable "firsts" during the early days of space race, and the public behind the Iron Curtain was made well aware of them (almost always after the fact), because they were seen as having immense propaganda value during the Cold War and were used to show domestically and globally the strength of the Soviet people, their economy, and their technology. There was certainly a lot of censorship of details (and total coverups as well), but it was in the USSR's best interest, both domestically and internationally, to publicize their space achievements.

In the satellite realm, some of the Soviet achievements were really monumental, and the basics of those missions were well known by the Soviet people. As examples, Sputnik 1 of course was the first satellite to orbit the Earth. Sputnik 2 was the first to send a living creature into space (the mutt Laika, who died on orbit). Sputnik 5 was the first to send living creatures on orbit and then return them safely to Earth. Their "Luna" series of probes also achieved some notable firsts in early lunar exploration, as well as some impressive early interplanetary exploration achievements.

Vostok 1 was the very first manned spaceflight and carried the first man to orbit the Earth, piloted by military officer Yuri Gagarin, who of course became a hero of the Soviet Union and known around the world for that achievement. After American Alan Shepard's short 15 minute sub-orbital flight shortly thereafter (America's first man into space), the USSR delighted in making it known to their people and the world that they had been the first to send a human up there, and orbit him, whereas Shepard just did a quick sub-orbital hop.

Vostok 6 carried the first woman into space - Valentina Tereshkova - and that mission was also highly publicized in the Soviet Union afterwards, as it was touted as showing the power of Soviet women in particular (especially since the Americans had no interest at all in allowing women into the astronaut program). Valentina's mission had some major problems, but those issues were kept from the Soviet public because they did not fit the positive propaganda narrative. Those issues were also known by American intelligence, who had listened in on her on-orbit radio comms from various global "listening posts" monitoring her flight, but the US intel community decided it was not worth really publicizing those points, as it would be looked at as "sour grapes" and might give away details of the US capabilities to monitor Soviet space activities (this was also the case on several other Soviet flights that experienced problems as well).

In regards to Apollo 11, here is what Nikita Khrushchev said of the topic - "Of course, you cannot have people land on the moon and just say nothing. It was published in all the newspapers. But if you remember [back then] when Americans spoke of the first man in space, they were always talking of "the first American in space" [not Yuri Gagarin]. The same feeling was prevalent in Russia. There were small articles when Apollo 11 was launched. Actually, there was a small article on the first page of Pravda and then three columns on page five."

So, the Soviet people were made aware by their government of many space activities by both their own teams and by the Americans, and were not kept entirely in the dark, but of course the US achievements were downplayed very heavily in the state-controlled press. By the time of Apollo 11's success, it was generally felt that the "space race" was over, and that the Americans had won. The Soviets then set their sights more on space station development in low-Earth orbit than possible lunar exploration, because they felt that was an area they could compete against the US in.