Well... I'm originally from Russia, but... not from the Cold-War era
But according to my Mum it was quite hard to come by Beatles albums in shops, so you would get them from people coming from overseas to Russia. For example, international students used to sell Beatles records to Russian students. Stuff like that.
Also, there's the Paul McCartney Live at the Red Square DVD, where people talk about the Beatles and Russia. Back in the days, you could get in trouble if you were caught listening to to the Beatles. Also, Band on the Run was banned for a while.
But still, the Beatles were much loved in Russia. My Mum used to listen to them all the time as a girl... even while doing her homework!
And that's lucky, coz K_E and I would probably never have become Beatles fans if it wasn't for her introducing us to their music