Ultimately, music is a medium in itself, its not a history book. However, I think that knowledge of the history behind the music can enhance the listener's experience. It can depend on the listener's interests, for example I've always been interested in history.
A big part of Shostakovich's uniqueness is that he bears witness to history. If there's one composer of the 20th century where its virtually impossible to avoid the circumstances in which the music was created, its him. I wouldn't go so far as say its undesirable to avoid the context, but I think that his music has added value exactly because it reaches out to the world beyond the concert hall. It addresses issues and explores areas that aren't easy to explore, even today with the events which inspired them being way in the past.
If listeners wish to venture deeply into his music, they can. When I first listened to Symphony #13 'Babi Yar' I didn't pick up the subtleties, because I wasn't aware of the history. I did some research on what actually happened at Babi Yar - the Shoah Foundation has done a lot in capturing eyewitness accounts by survivors on video - and then I realised why its such a significant work. With its fusion of music and poetry, its a memorial in music that demonstrates how Shostakovich was a master of narrative.
Ironically, while being accused of it by Zhdanov in 1948, Shostakovich went against the grain of formalism which was in vogue in the West. I doubt that his approach to music would have been much different had he moved to the West. It might have been worse, for example if Prokofiev had stayed in the USA, he would have been restricted to the life of a touring virtuoso like Rachmaninov. His own music would have taken a back seat, and American audiences didn't react well to even comparatively accessible works like Piano Concerto #3.
That's another side to this discussion, that we needn't buy into the Cold War dichotomy - from the Western viewpoint, at least - that East=stagnation, West=progress. Everyone focuses on Stalin, but Khrushchev did a lot to liberalise the arts. The Zhdanov Decree was rescinded in 1958, which opened up the path to cultural exchange - Russian musicians could work in the West, and Western musicians could do the same in Russia. Poland, one of the Soviet satellites, became a centre of new music in Europe with the Warsaw Autumn Festival. There are many sides to Shostakovich's story.