Production started during an era that was a slight relaxing of censorship in general in the USSR. Obviously any criticism of the Soviet system, law, government, policy, leaders, etc. was absolutely right out, but you do start to see hints of social criticism in that era. There are a rash of novels in the late 50s and 60s in that style.
In fact, just the year before, Krushchev had personally ordered the publication of One Day In the Life of Ivan Denisovich, which is also quite hard to believe got published in Soviet times, even with redactions. The leash was pulled short again not long after that!
Anyway, legend has it that Krushchev also got a preview of "Welcome, or No Trespassing" and liked it. So he ordered it immediately released. I don't know if that's true, but it certainly could be.