The news of the, for example, large-scale deportations of ethnic groups that Stalin didn't like was, unless I read it wrong, generally airbrushed into something a bit more palatable in the press.
The show-trials and purges were pretty much trumpetted in the press. The aim seems to have been to increase the paranoia - show how insidious the 'revanchist' or 'counter-revolutionary' movement was so that all party members outdo themselves proving that they aren't one of them - usually by pointing the finger at others. Stalin was great at 'divide and conquer' amongst anyone he saw as even a potential threat. He was in the paranoid, megalomaniac's dilemna of needing to assert his complete dominance over his closest advisers and to stifle any independence of thought whilst requiring talent and vision from them.
It appears that the only people he even remotely trusted were some fellow Georgians/Minghrelians such as Beria, although he was probably more tolerated due to his efficiency as an agent of state terror rather than trusted, and some of Stalin's civil war buddies like the mustachioed pillock, Semyon Budyonniy and also Klimenti Voroshilov.
Another thing about Stalin is, to paraphrase Douglas Adams, he wasn't above a bit of vicious anti-semitism in the same way that the sea isn't above the sky. The Show-trials and various other staged events at that time were aimed at old Bolsheviks who were not in Russia during the civil war - and also the so-called 'International Bolsheviks' such as Kamenev, Zinoviev and so on. They were all Jews. As for what happened to Trotsky, another Jew - that is pretty well-known.
I doubt that the quotas that regional NKVD bureaus were given for finding dissidents and the enthusiastic measures that were often taken to beat those quotas were widely advertised. Certainly the death of the disgusting Yagoda, who oversaw much of this, came several years too late for many, many innocent Russians.
For a pretty good introduction to the twisted workings of the Stalin regime, you could do worse than read "The Court of the Red Tsar" by Simon Sebag-Montefiore or, if you can see past the self-serving crap, Sergo Beria's biography of his dad, "Beria".
If you ask Russians today who were alive during the 30's what it was like, many of them have very nostalgic recollections of a country that was developing and industrialising at a very, very fast rate and they are quite proud of that fact. They also tend to talk in rather vague terms of 'counter-revolutionary elements' that were dealt with, adopting the attitude that you couldn't make the omelette without breaking a few eggs. In this respect I would say that there definitely was quite a good cover-up.
Of course, such people are looking back to the days when they were young and so are very often wearing rose-tinted spectacles.