You are asking multiple questions here:

Were there … everyday citizens of the … Eastern Bloc who believed in communism?

If so, how many ?

Were there more people supporting communism in the Eastern Bloc or in the West ?

These questions are incredibly hard, if not outright impossible to answer correctly.

 

Communist era

Keep in mind all communist regimes in the Eastern Bloc were dictatorships; publicly not supporting communism was not an option (one could argue it was dangerous to do so even in private, thanks to the informer system). Therefore trustworthy contemporary „surveys”, if you will, for the support of communism do not exist for this reason.

You can take into account the membership numbers of communist parties in each country (they were roughly around 10% of the population in most of the Eastern Bloc); but this again wouldn’t give a true picture, as entering the party was often times motivated by gaining better career options and ambition, and not belief. [1]

Perhaps the only true "indicator" you could gain from this era is the presence of multiple revolutions in the region. These were only quelled by the military interventions of the Soviet Union. These events suggest the group you are looking for („believers”) was a minority in the whole society.

 

Post-communist era

The next step would be to the check right after the collapse of the communist regimes in the Eastern Bloc. However, here you also wouldn’t get a true picture, as even people who sincerely believed in the communist ideology, during questioning would hide it at that time, for obvious reasons.

Perhaps next you can ask, what about the first democratic elections? Multiple communist parties had successor parties running in the first elections. The democratic elections are anonymous, so this fear of retribution would be less of a factor.

These first democratic elections turned out to be "by and large electoral disasters for ex-communist parties" [2], most of them hardly got 10% of the votes.

 

And just for the sake of it, we can go even one step further, to the next democratic elections half a decade later, where the initial hostile „anti-communist” climate had pacified a bit. What happened was that "the renamed and reformed communist parties staged a dramatic comeback in national and state elections across the former communist world."[2]

A good example is Hungary, where the successor party got the majority of the votes and was able to form a government. Unfortunately, this is yet another false indicator, as it has been shown, that "the more difficult a population perceives the process of economic transition, the greater the electoral success of an ex-communist party." [3]

 

The West, Summary

I'm only going to briefly mention the West here. The largest communist support in the Western Bloc was in Italy, where the Italian Communist Party numbered more than a million members in the 2nd part of the 20th century, and gathered one fourth to one third of the votes in multiple elections.

 

To summarize, we do not know for sure how many everyday citizens truly believed in communism in the Eastern Bloc. From the presence of revolutions we can speculate it was a minority rather than a majority. But there was indeed a portion of society who believed in it, as evidenced by the vote numbers of successor parties.

 

Sources:

[1] <IN HUNGARIAN> Hell Roland (2010): Additions to examination of the elite of the Kádár-regime: the composition of the membership and independent apparatus of the HSWP reflected in numbers. Jog, állam, politika. (for further reading: link)

[2] Ziblatt, Daniel (1998): The Adaptation of Ex-Communist Parties to Post-Communist East Central Europe: a Comparative Study of the East German and Hungarian Ex-Communist Parties. Communist and Post-Communist Studies. (for further reading: link)

[3] Evans, G. and Whitefield, S. (1995): Economic ideology and political success: communist-successor parties in the Czech Republic, Slovakia and Hungary compared. Party Politics.