There were several instances of cooperation during the space-race; Russia was not entirely isolated. Russia (USSR, to be precise) was an active participant in the IAF, and UN programmes such as COSPAR.
To quote Brian Harvey from "Rebirth of the Russian Space Program"
The Soviet Union ran cooperative programs with the United States from 1965, especially in space biology and together the two countries hosted annual conferences on lunar and planetary exploration
You'll find more about the complex relationship in the space arena between the mentioned protagonists in that book, and the link provided by Noordung.
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