I answered a bit about Americans moving to the USSR in the 1930s in two threads, here and here. There is always more to say on that particular subject alone, but that is a good starter especially in terms of English-language histories on that subject.
However, that is all pre-Cold War. There were definitely cases of defections to the communist bloc in that period - Lee Harvey Oswald moving to the USSR comes to mind - as well as some prominent, extended public tours, such as Angela Davis in the 1970s. Personally I'm not sure anyone has tried to write a comprehensive history on the subject of defections from the West. It's also worth pointing out that even using the term "defector" raises particular issues, because it tends to lump together people who were ideologically supportive of the communist bloc, people who maybe did not have strong feelings about the communist bloc but were hostile to the Western bloc, and people who were intelligence assets of Eastern bloc intelligence agencies. So for example someone like Oswald moved to the USSR for ideological reasons, but someone like Kim Philby defected because he was a double agent for the KGB.
I can only speak about Bertolt Brecht specifically. He may not quite be what you're looking for since he was originally born in Germany, but he may be of some interest. He had attained some reputation as a playwright and poet before fleeing the Germany (his work had a decidedly Marxist bent, though it should be noted that he would later deny ever being a member of the communist party) shortly after Hitler came to power. Having lived in various other European locations (during which time he would write some of the plays for which he would become most famous, e.g. Mother Courage and her Children, and The Life of Galileo), he eventually received an American visa in 1941.
Brecht continued to work in the states, writing among other things The Caucasian Chalk Circle. His plays continued to be politically challenging; while perhaps stopping short of explicitly endorsing communism they consistently encouraged the audience to ask challenging political questions and to think for themselves. Brecht was brought before the Committee for Un-American Activities in 1947, and testified that he had never been a member of the Communist party. Brecht found himself blacklisted by movie bosses and returned to Europe.
Now, my supervisor at one time asserted that Brecht desired to return directly to Communist East Germany, but that the powers at the time viewed his political plays to be, while marxist, somewhat too radical, too politically challenging, too much encouraging the people to think for themselves - more a troublemaker than a party loyalist. I have frankly not got a source to back this up, but it rings true in that after leaving America (the day after he testified to the Un-American Activities Committee) in October 1947 he did not immediately return to East Germany, but rather to Zurich in Switzerland. He was eventually allowed to take up residence in East Berlin in 1949 and lived there until he died in 1956. He continued to write, but his later plays did not attain the fame of his work while in exile and he remained important mainly as a director. Brecht accepted the Stalin Peace Prize in 1954.
Brecht died in East Germany in 1956 (at the age of 58) and is remembered as one of the most important 20th Century German playwrights and poets. I've visited his grave in East Berlin; it's a fairly plain grave in the Dorotheenstadt cemetery in Berlin Mitte. The same cemetery contains a monument to Dietrich Bonhoeffer and the graves of number of other Nazi-resistors and other historical figures with whom I am sorry to say I was not familiar.
So yeah, Brecht desired to return to the Communist DDR rather than stay in the USA, was not immediately welcomed, but managed to play his cards right and to do fine until his death, if not managing to gain the same artistic acclaim that he had enjoyed in exile.