This is an overarching question of a number of interrelated questions I've been curious about- did Russian citizens, and after WW2 people in satellite states, manage to gain information/ put out information around the censors and if so how? Did new technology help or hinder?
I can provide a short response in reference to the Chernobyl disaster. One of my professors was in the USSR at the time of the explosion (April 1986). He explained that government censorship meant that he did not hear about the disaster until after he returned to the US after the accident but before the official Soviet response.
Initially the Soviet government denied or downplayed Western reports of elevated radiation levels across Scandinavia. Ultimately the Kremlin released an official and full account of the disaster a week later.
Prior to that, citizens of the USSR were officially in the dark. Exceptions existed for those in Pripyat, the town where the Chernobyl workers lived, and East Germany. Many people in the GDR could receive West German television signals and were fully aware of the dangers but were reassured by East German media that radiation levels were normal and they were not at risk.
Mind you this is 1986, toward the end of the Soviet Union and the government still engaged in censorship. I would say that technology and proximity to the West meant that many people could subvert censorship if they were lucky or determined enough to gain access to Western information.
These articles provide a good read about the official response to the disaster in both the GDR and the USSR as a whole.
http://www.spiegel.de/international/east-germany-and-chernobyl-the-censorship-of-fear-a-412211.html