First, I'd like to point out that the Soviet satellite states may not have been “puppet states” in the way you mean. The Soviet could and did use the various levers that provided it with hegemonic power in Eastern Europe (political influence, the secret police, military power, etc.) to directly influence the internal politics of the Eastern Bloc and even to remove undesirable politicians from positions of power. But the Soviet Union did not dictate the internal politics of these nations and in fact had to approach them as independent nations capable of pursuing policies that challenged Soviet hegemony in certain ways. The best example of this system was the rise, fall, and reemergence of Polish leader Wlydyslaw Gomulka. Gomulka rose to the top of the Polish Communist Party during the Second World War, and thus to the top of the Polish political system after the war. Stalin, however, favored the Moscow-bred Boleslaw Bierut, and Gomulka was overthrown and imprisoned. When Poland fell into chaos during the anti-Soviet wave of 1956, Gomulka returned to power and the Soviets were forced to accept both his leadership and his “Polish Road to Socialism,” however. The Soviet Union could control Polish politics to a point, but its leaders also recognized that they faced limits in maintaining the Easter Bloc (this is but one example, as the Soviets also conceded reforms elsewhere, most notably Czechoslovakia in 1968 until things got out of hand, and practically lost control of Romania around the same time – this is off topic, so I'll get to the question at hand now).

As for the question at hand: yes, Eastern Bloc citizens were very aware of the hegemonic power of the Soviet Union and their own nations' subservience to Moscow. There really is no question about this. It was widely known since the Yalta Conference of 1945 that Stalin had demanded the creation of a “Soviet sphere” after the war and that the Western powers had conceded to some extent, and after the war Moscow forced Stalinist governments on most of the Eastern Bloc (Czechoslovakia, which actually voted for a Communist-led government in 1946, was an exception at the time). The East German workers waged a mass protest against Communist/Soviet control in 1953, and the East German government with the assistance of Soviet tanks put down the movement. Then, in 1956, Hungary waged an open rebellion against the Soviet Union and tried to leave to the Warsaw Pact (the military alliance of the Eastern Bloc) altogether, but the Soviet Union crushed the rebellion by force. That same year, Poland nearly rebelled, but, as noted above, the Soviet Union compromised and allowed the domestically popular Gomulka to seize control in Warsaw. In 1968, political reforms in Czechoslovakia allowed, among other things, for a free press, and Czechoslovakia writers as well as ordinary citizens wrote and published extensively about the Eastern Bloc's subordinate position to the Soviet Union, and this helped to spawn student protest movements elsewhere, notably in Poland. To quell this discourse, the Soviet Union again invaded, this time to put down Czechoslovakia's “Prague Spring.” Millions of Eastern Bloc citizens understood the implications – the Soviet Union would not relinquish its hegemonic position within the Eastern Bloc.

Tl;dr – The Soviet Union did not secretly control Eastern Bloc politics, but rather overtly enforced its hegemonic position at the top of the Eastern Bloc alliance, leaving the citizens of the Eastern Bloc, who occasionally resisted Soviet hegemony, with a clear understanding of Soviet power.