(NOTE: contains brief NSFW reference.)

The actor was receiving a visit from a star of the Bulgarian theatre. This visitor said to me, "You know, Streetcar Named Desire is being done in Moscow and Sophia now." I told him I’d heard about that and had also heard that Blanche did not go mad over there. "She goes mad in Bulgaria," he told me, "but not in Moscow." And I thought to myself, "Those Moscow cats must have a lot on the ball to keep Blanche in her right mind."

Foreign literature was quite popular; for example, in the period between 1958 and 1959, the editor of the journal Inostrannaia literatura (which specialized in foreign literature) claimed that 54 million books by foreign authors were published in the Soviet Union.

As you intuit, some books were available and some were not, but let's first list authors sorted by popularity, as published from 1918-1957:

Jack London, Victor Hugo, Jules Verne, Hans Christian Andersen, Honore de Balzac, Mark Twain, Theodore Dreiser, Guy de Maupassant, Emile Zola, Charles Dickens, H. G. Wells, Romain Rolland

You'll notice a penchant for social progressivism, although in some cases -- like Jack London -- authors were popular even before the Revolution.

There was certainly consideration of which authors were "correct" to publish, in a political sense. The fortunes of John dos Passos are a good example.

He was an American novelist considered during the early 20th century to be at par with the finest. After World War I he acquired a strong bent towards socialism. He was one of the multitudes of foreigners who supported the Republicans in the Spanish Civil War, but a visit to Spain in 1938 led him to be less enamored with the left to the point he wrapped all the way around to the conservative right and campaigned for both Nixon and Barry Goldwater.

His most well-regarded set of works is his U.S.A. trilogy:

The 42nd Parallel (1930)

1919 (1932)

The Big Money (1936)

The first two were translated to Russian and published, as John dos Passos was socialist; but when he changed -- shortly after the English version of The Big Money -- Soviet publishing of his works stopped (eventually, a decade after he was dead, the first two books of the trilogy were republished).

Other hot authors included Tennessee Williams, Eugene O'Neill, and James Baldwin (black authors were fairly well received in general). As the quote on top indicates, their reception may still not be without editing.

To consider a somewhat apolitical change of literature -- albeit one that still has resonance today -- is Baum's book The Wizard of Oz. It was "re-imagined" in 1939 by the author Volkov as The Wizard of the Emerald City, and it essentially a different book: Dorothy Gale becomes Элли Смит (Elli Smit) who rather than being an orphan lives with her mother and father. Part of the prologue is from the perspective of the Wicked Witch of the East (rather, Гингема, "Gingema"). Toto talks. Subplots are added and changed.

Volkov wrote five sequels, entirely independent of Baum's plots, and the books are better known in some countries than Baum's originals are.

Censorship in translation for political reasons most certainly did happen, sometimes blatant, sometimes subtle. A training textbook for the editors of Inostrannaia literatura notes that

It is indisputable, for example, that the editor not only has the right, but the duty, to demand that the contents of the manuscript meet the interests of the Soviet state ...

Consider, as a specific example, the novel Meeting at a Far Meridian (1962, Mitchell Wilson), about an American physicist who goes to Russia to work with a Russian scientist. Some of the editors found the work "interesting" and promotional of peace between the countries; others were worried about the romantic interest (a Russian, Valia) having an affair with the American. Others thought the work would "equate the two worlds", and the editors agreed that an aspect that "shared blame" for the Cold War could not be included.

... in the novel there are a number of moments which it would be desirable to remove or soften in translation.

In addition to politics, sexuality was particularly hedged around and edited out. Consider the opening of Catch-22 (1961, Joseph Heller):

It was love at first sight.

The first time Yossarian saw the chaplain he fell madly in love with him.

When translated in 1967, these lines were omitted entirely, any possible homosexual implications swept away. Almost unbelievably, Breakfast of Champions (1973, Vonnegut) has a Cold War translation, meaning lines like

He had a penis eight hundred miles long and two hundred and ten miles in diameter, but practically all of it was in the fourth dimension.

never made it to the translation.

The general claim from Soviet critics of Americans is that the non-socialist authors were "blinded" by their local politics. Returning to Tennessee Williams -- arguably the most popular American playwright in the USSR during the Cold War -- even he was considered flawed, as while he wanted justice, he also -- according to the Soviet critics -- considered it to be "beyond humanity's reach". Perhaps this is why the Moscow version of A Streetcar Named Desire changed the plot to have a happy ending.

...

Chulanova, T. (2020). The Politics of Paratexts: Framing Translations in the Soviet Journal Inostrannaia Literatura (Doctoral dissertation, Kent State University).

Friedberg, M. (1976). The U.S. in the U.S.S.R.: American Literature through the Filter of Recent Soviet Publishing and Criticism. Critical Inquiry, 2(3), 519-583.

Koch, S. (2006). The Breaking Point: Hemmingway, Dos Passos, and the Murder of Jose Robles. Robson.

Khmelnitsky, M. (2015). Sex, Lies, and Red Tape: Ideological and Political Barriers in Soviet Translation of Cold War American Satire, 1964-1988 (Doctoral dissertation, University of Calgary).

Ruggles, M. (1961). American Books in Soviet Publishing. Slavic Review, 20(3), 419-435.

Shabad, Theodore. (31 August 1982). American Writers Seen Through a Soviet Glass. The New York Times.

Sherry, S. (2013). Better Something Than Nothing: The Editors and Translators of Inostrannaia literatura as Censorial Agents. The Slavonic and East European Review, 91(4), 731-758.

Slavova, K. (2014). Tennessee Williams on the Bulgarian Stage: Cold War Politics and Politics of Reception. DQR Studies in Literature, 54, 235.