To answer this rather complex question at least in part - that is for the specific case of Czechoslovakia mentioned as an example - we first need to define certain terms so it is clear what we are talking about.

As this is a matter of politics and public opinion from a recent era that is still in living memory, it is necessary to understand that opinions of people may vary a lot. For that reason, I will try to be as objective as possible, but keep in mind that someone with a different viewpoint might focus on completely different aspects of this issue and thus reach different conclusion.

Let's define the terms:

We are talking about the Eastern Bloc and I will use Czechoslovakia as an example. The official ideology in Czechoslovakia from 1948 to 1989 was Marxism-Leninism. This ideology defines socialism as public or also collective ownership of means of production and communism as public or collective ownership of all property. Or in terms used in the constitution, socialism is based on the statement "Everyone according to his abilities, to everyone according to his work" and communism is based on the statement "Everyone according to his abilities, to everyone according to his needs" https://www.psp.cz/docs/texts/constitution_1960.html

Even though we are talking about the Eastern Bloc, the term "eastern Europe" is not really proper for regions such as the one of Czechoslovakia, DDR or post-war Poland as they are part of Central Europe and are historically lands belonging to the Latin west. As this difference is generally seen as important by many people living in those countries, when debating them, the term Eastern Bloc is preferable. It includes the countries of the second world, including for example Cuba.

For simplicity, I will only talk about the official economy, not the black market, which of course existed, as everywhere else, as illegal free market economy outside of the power of the government.

Now, let's look at what Czechoslovak government looked like. Czechoslovakia was established in 1918 as a constitutional republic, a parliamentary democracy and it's economic system was based on relatively free market with certain social security aspects, typical for countries such as France or the United Kingdom. The Czechoslovak communist party was founded in 1921, became part of the Comintern, but didn't play a major role in the state affairs of Czechoslovakia until the second half of the 1930s when it temporarily tolerated a centrist government.

During WWII, unlike the Czechoslovak government which went to exile in London, the communist party leaders and active members went to exile in Moscow. They started active resistance and became official part of the resistance movement in 1941 when the USSR went to war with the Third Reich (Luža R., V Hitlerově objetí, str. 55, ) Even though Czechoslovakia was liberated by several armies, the leading role in the public perception was the one of the USSR, be it the Soviet army itself or Czechoslovak soldiers serving in it.

The president and the government in London exile made an agreement with Moscow and nominated a new Government at the end of the war where the Czechoslovak and Slovak communist parties had the highest number of ministers and a dominant role. In 1945 elections to a Temporary National Assembly, with a limited amount of political parties, the communists gained around 1/3 of the seats in total. Their position was improved in 1946 elections and Klement Gottwald, chairman of the Czechoslovak communist party became the prime minister.

All this development helped to move communists from an extremist anti-state entity to a position at the top of the government and openly Stalinist Gottwald, who once told members of the parliament "We are going to Moscow to learn from the Russian bolsheviks how to twist your necks." became arguably the most powerful politician. https://www.psp.cz/eknih/1929ns/ps/stenprot/007schuz/s007003.htm

So far, the Communist party gained democratic legitimacy and started the first two-year plan as a step into planned economy, yet with respect to private ownership of small businesses. The economy was already heavily under state control from the time of the nazi economic changes and big business was either fully nationalised or heavily under state control.

In 1948 the communist party through deals with large part of social democrats and former independents organised a takeover of the entire government (last non-communist minister was found dead few weeks later) and February of 1948 is considered the actual start of the rule of the Czechoslovak Communist Party, even though other smaller parties remained in the National Front coalition.

The declared goal was clear from that point and it was following the Soviet template in all aspects of life. From politics and economy to teaching communist ideology social values to every single person.

(end of part one)

In the case of Czechoslovakia, it was essentially a political statement that accompanied the passage of a new constitution in 1960. The beginning of the 1960 Constitution (which you can read in English here) provides a little potted history of the country since 1945, concluding that the period of "people's democracy" was over and capitalist exploitation had been eliminated in Czechoslovakia. Materially speaking, what this meant was basically that the means of production were no longer in private hands (although collective farms had not been established everywhere by 1960, the most productive land had been collectivized, as had nearly all private businesses). Symbolically, it also served to mark the official end of what Communist officials understood as the struggle with reaction and internal enemies and the establishment of a united socialist society, and what we might understand as a period of internal bureaucratic chaos, infighting, and political inconsistency. The 1960 constitution also named the Czechoslovak Communist Party the "guiding force in society". Communist leaders had understood themselves as such for years, but this made it official.

The rhetoric surrounding the passage of the 1960 Constitution was that both Czechoslovakia's path to communism and therefore the well-being and prosperity of its citizens was secure. This was a period when officials were really trying to demonstrate to the population that the socialist economic system could secure them a better standard of living than any other, and when this seemed like more of a reasonable proposition as well.