There were substantial efforts by the US to infiltrate the Soviet missile development complex in the 1950s and early 1960s, when the Soviets were ahead in the Space Race, but these efforts were spectacularly unsuccessful. The Soviets had 'closed cities' for the missile and nuclear industries which isolated their inhabitants from the general population and were frequently geographically isolated as well, this meant that on the technical level there were very few opportunities for industrial espionage. The extent of the Soviet blockade on technical data was sufficiently extensive that the US didn't even find out the names of those working on the technical side of missile design. Sergei Korolev was known simply as 'The Designer'.

There was always some data gathered by EMINT (Emissions Intelligence) operations by the US using spyplanes and the early spy-satellite program, when the Soviets were testing, but in general they managed a high level of secrecy. In part this led to the US myths of first the 'Bomber Gap' in the 1950s and then the 'Missile Gap' around 1960.

In other less sensitive areas of production there was less focus on espionage as the US had been, and remained, ahead in consumer technologies through the Cold War.

Source: Cadbury, D., The Space Race: The Battle to Rules the Heavens

Wolfe, T., The Right Stuff