I suspect that there's a different answer depending on the country you're looking at. I don't much like the term "satellite state," not because I deny that the Warsaw Pact countries were under Soviet hegemony (I don't), but because it suggests that their own internal dynamics aren't relevant for explaining their political development.

Anyway, I can provide something of an answer for Czechoslovakia. I should first mention that Czechoslovakia was not occupied after 1945, and Communist dictatorship wasn't established there until February 1948. Because the Czechoslovak Communist Party was fully legal and participated in elections throughout the interwar period (this wasn't the case in most other East Central European countries) it had an established party infrastructure and popular base of support. The postwar party was led, more or less, by the same people who had led it since 1929, when a faction of younger, more radical functionaries overthrew the existing party leadership and brought the Czechoslovak Communist Party closely in line with the Comintern. That is to say, the party's leaders were very much both home-grown Communists and Soviet loyalists.

It sounds like when you wrote "Communist party members" you were referring to these leaders rather than ordinary members, but it's worth noting as well that the Czechoslovak Communist Party grew massively after the war. Just before the war, they had had around 70,000 members, but by 1946, they had over a million -- and by 1948, over two million (out of a population of about twelve to thirteen million at the time). The Communists won about a third of the vote in the 1946 elections (which, although not what we would call totally free, were not rigged in their favor). Many people had a favorable opinion of the Communists because of their prominence in the resistance to Nazi occupation, because of the Soviets' role in liberating Czechoslovakia and defeating Germany, and because of what they saw as the failures of the interwar liberal democratic system (not just its inability to defend the country from Nazi aggression, but also persistent poverty and inequality, exacerbated by the Great Depression). The Communist Party took advantage of this position by promising a "Czechoslovak road to socialism" that would fulfill Czechs' and Slovaks' aspirations for national sovereignty and security as well as social and economic justice, and would be based on popular participation at the local level rather than centralized bureaucratic control, and allow for the existence of certain other political parties as well. (For anyone who might think this sounds appealing, bear in mind that one of the first stops on this road was the ethnic cleansing of the German population, it didn't permit ordinary parliamentary political competition, and it came with a huge amount of paranoia about and denunciations of suspected reactionary elements and former fascist collaborators.) This didn't initially conflict with Soviet strategy for East Central Europe. As the Cold War started heating up, and particularly after Yugoslavia was kicked out of the Comintern, however, the Czechoslovak Communists were forced to get off the national road and travel down the Soviet one. It was in this context that the Czechoslovak Communists took advantage of a political crisis to seize power in the government and establish one-party rule.

I can't really speak to the relationship between Moscow and other Eastern European Communist parties, but I imagine that it would have been very different in a country like Romania, whose Communist party was both tiny and illegal during the interwar period, or Poland, whose Communist party was actually dissolved and many of its leaders imprisoned or killed because of Stalinist paranoia about Polish spies.