Unfortunately this is quite a difficult question to answer - mostly because the definite starting point for "living under communism" is incredibly vague when it comes to East Germany, and it is also not like we can go look at the opinion polling. So there will need to be some approximation here - which will necessarily have to touch on the pre-war period and go a bit into the 1950s.

Germany had a thriving communist movement which staged several failed revolts throughout the 1920s, and despite these failures and quite severe repression, still maintained millions of supporters. The rise of Nazism went hand in hand with the smashing of left wing political parties, trade unions etc. and therefore that support was unable to take expression, even if there were isolated cases - the resistance movement in Germany generally etc. This is important because it shows that despite the vicious anti-communism of the Nazi party and the repression of the communist movement, the support for communism in Germany as a whole was not dead during their rule.

Essentially, much like the rest of the Nazi ideology, the ideological anti-communism of the Nazi party was not necessarily entrenched in German consciousness insomuch as people just accepted it as part of their overall capitulation of fascism. And once the Nazis were defeated, those who accepted it adapted to the new status quo.

But it is worth thinking about the new status quo for a moment. Was it "living under communism"? Ignoring completely the intraleft debates about the nature of communism and if the Eastern Bloc was "real communism" which are certainly debates that would have been live at that period in time, the actual establishment of communist rule in East Germany was relatively gradual and obscured behind numerous layers of political theatre. For example, rather than the Communist party of Germany taking power, there was a (forced) fusion between the Communists and Social Democrats (who had previously been killing communists prior to the rise of Nazism) into the Socialist Unity party (SED). But even so the SED was not the sole ruling party - it governed as part of a (completely superficial) coalition first known as the Democratic Bloc and then later the National Front.

Both of these coalitions included parties across the political spectrum, and every elected representative was a member. All were ultimately subordinate to the SED, but not without some difficulty - the SED did not secure full control until the tail end of the 50s, and it involved purges and imprisonments both within the SED (against more independent and democracy-minded communists) and externally in other political parties of those who were not happy with the emerging status quo.

Ignoring the political side of things for a moment, the Soviet social and economic system was never forced on East Germany in the same manner as it was in the other Eastern Bloc states. This is most likely down to the partitioned nature of the state at the time and the fact that the Soviet leadership still held out hopes for a unified and neutral Germany until the NATO powers rebuffed it in 1952.

Up to this point, East German living standards had generally been improving relative to the absolute lowest points during the immediate post-war period, and so there was not too much popular discontent. However, the failure of Soviet-brokered reunification in 1952 caused the East German leadership to embark on "Sovietisation" - basically a programme of transposing the Soviet cultural, social, economic and political systems onto other states. Sovietisation enjoyed varying degrees of success across the Soviet sphere but was disastrous in East Germany. Wide spread capital strikes by the remaining private industry in response to nationalisations and discriminatory taxation, and wealthy farmers refused to integrate with the state collectivisation plans and left the country, resulting in a major contraction in food production as much of the land they worked lay fallow. Austerity measures were introduced as a result of the dire economic situation, coupled with higher production quotas and wage cuts - in short, state welfare was cut, you had to work more and you had to work for less while you were at it, and all the food prices just went up, if there was food at all. All this occurred in the space of a year.

Stalin died in 1953 and the new Soviet leadership took power and immediately recognised that the East German government had created a big problem - ironically, the Soviets thought that the East Germans were too zealous and had gone too far in their Sovietisation, and stepped in to advise the East German leadership to correct course.

Unfortunately for everybody involved, the situation had already deteriorated too far and it resulted in an uprising in the second half of 1953. The straw that broke the camel's back was a 10% increase in work quotas without any increase in pay - workers went on strike and it culminated in mass demonstrations of millions against the government. Despite whatever criticisms the Soviet leadership had for the SED, they were determined to back them, and both the East German government and Soviet military repressed the demonstrations. The result of the repression was the shoring up of the regime but a complete shattering of illusions in East Germany's new communist government. Within the SED itself, there was huge discontent in the ranks of the party and a mass exodus of members, and the leadership of the SED was completely discredited within the party, only kept in place by the backing of Moscow. The Sovietisation reforms were stalled and the East German leadership committed itself to more gradual changes in the future.

So, what did East German people think when they realised they had to live under communism? If we take the reaction to Sovietisation, they were not happy - but that had essentially nothing to do with the legacy of Nazism and its ideology of anti-communism, and more everything to do with the rapid deterioration of their material living conditions in the 1950s. But prior to that, it would probably be a bit much to say that East Germans were "happy" living under the government imposed by the Soviet Union, but they at the least were not up in arms about it and were willing to give it a chance.