I'll talk about suicide in Hungary since the country got much attention in the 80s due to reaching a sad world record rate of 45,1 per 100 thousand people in 1987 and because of the (in)famous Gloomy Sunday song, marking a tragic coincidence that made people speculate about causation and prompting serious research. I'll rely on study done by Central Statistical Office and the work of Bela Buda, a renowned Hungarian psychiatrist specialising in suicide prevention.

Before that, quick answer: The GDR kept the data, just didn't make it public.

According to these sources the developed world's suicide rate fluctuated between 5-15 per 100,000 from the 50s to 80s (varies lot by country). Compared to them, as you can follow on the chart even during the relatively prosperous 60s suicide rates were above 25 and steadily climbing until hitting a high in the mid-80s. Now behind this numbers there were vast regional, age- gender-, family- etc variations, though divorced males in rural settings having a clear majority. The main conclusion of the Hungarian statistical study is that suicides are best explained as a result of economic-social settings creating peculiar cultural spaces that either encourage or discourage suicide, or in other words how acceptable suicide as an option is. Dr. Buda attempts to provide a further cultural framework: In his view suicide has long been a trait of Hungarian culture, with many famous historical persons opting for this path, or in his words: ""[the culture] favouring defective, maladaptative patterns of solution for life problems". He quotes interviews with suicide helpline talkers who tell them about how people view it as an easy way out. Beyond suicide, Hungary as a whole suffered many other serious mental health problems in the time: Though I can't find the sources that backs up his claim that quarter of the population had dire anxiety, he and the Statistics Office both refer to the high rates lf alcoholism in the country, exemplified by the 7-8 thousand cases of liver sclerosis per year in the 80s (This statistics haven't improved much: the lead toxicologist of Hungary estimates 1 million serious alcoholists in the country, and a further 1-1,5 million [out of a total of 9,8 million), also shown by WHO Alcohol Status Reports).

So why the bad mental health? Buda says one potential reason was alienation: During the forced industrialisation, collectivisation and subsequent urbanisation of the 50s and 60s, nearly 60% of the country changed residence, breaking up traditional ties of kinship and communities. The daily realities of socialist economy aggravated the life of your average worker: After small market-oriented reforms were introduced from the late 60s, inequality began to grew as a relatively small group began to profit from the situation, while for most people work offered little future other than iddling away at some factory, often barely doing much work all day (as unemployment was illegal and worker shortages common, factories and other companies basically had to overlook that often serious share of employed barely contributed, also for managers and mid-level leaders there was little incentive to change the situation due to lack of competition and guaranteed state subsidies). This also helps explain the drop from the 90s: While interestingly the Statistics Office simply failed to find any strong causator in their models to explain why despite the sharp economic crash during the political-economical transformation suicide rates dropped. Buda proposes that despite the hardships, because hope and potential suddenly appeared from nothing following the political changes, suicide rates began to sink, though he does add that the general bad mental state of Hungary persisted.

These works don't touch much into other eastern block countries, but publications like this does indicate higher average suicide rates for Eastern Europe as whole, I'm hoping someone else will fill this in.

Sources:
Suicide in Hungary 1970-2010, Central Statistics Office, T. Zonda, B. Paksi, E. Veres
Suicide - a medical and sociological study, Dr. Bela Buda
WHO Global status report on alcohol and health 2014

I found an article about this very topic by a historian who has worked on suicide in the GDR. The suicide rate was indeed higher in the GDR than in West Germany, but this author attributes it to poor mental health treatment and politically motivated stigma around suicide rather than direct political repression, as the movie suggests. This review of his book (which is in German) also points out long-standing cultural factors that contributed to a higher suicide rate in the GDR (apparently there is a correlation between Protestantism and suicide, for example).