I reckon: not as much as we think we do.
Just been reading a story about President Obama, Russia, Crimea, Ukraine, sanctions, that stuff, on the Guardian website. When you read the comments section, it's apparent that a lot of them are written by Russians. It's quite striking in fact. They are apparently posted by "Putin-bots" working in Savushkina Street (Google it if you don't already know) although I'm not really sure I believe that. They're always very pro-Putin, pro-Russia and anti-West, often quite virulently.
The Western view of Russia is something like: Putin does what he likes and only the State version of events is reported to normal citizens. Funnily enough, this isn't a great deal different to what seems to be the Russian view of the West - Obama and the EU do what they like and through propaganda, convince normal citizens that their version of events is right.
Let's assume Putin is (although I doubt he's had a clinical diagnosis) insane (in a kind of Bond villain way). However, I do think he's genuine - as in he believes that what he's doing is right - he didn't "annex" the Crimea or make moves into Ukraine (and who knows where in the future) for some power-grab, he did it because he sees these territories as parts of Russia to be reclaimed. His always-high approval ratings and constant re-elections (unless all that's rigged) suggest that most Russian people agree with him, too. "Ah yes", they naysay, "but they only believe what they're told".
I'm not suggesting that the Russian view is right by any means, but I don't think their news is censored as much as we seem to think it is, especially for English-reading Russians. I just sometimes wonder if what we're being told is accurate, given that most of our media has an agenda of one sort or other.
Hopelessly out of my depth on this one, I think. You know sometimes you think you want to say something about something, then realise you don't really know what you're on about? You Are Here.