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For US citizens detained in USSR :
Legal frame
Since 1933, USA and USSR had established a protocol according to which each country has to inform the consul of the other one
whenever a national of the country which he represents is arrested in his district [or] if a prisoner is transfered from one place of detention to another.
It is mentionned at the highest state level, in this extract from correspondance between Roosevelt and Litvinov.
The case of Isaiah Oggins
The American communist Isaiah Oggins was arrested by the NKVD on February 20, 1939, after having spied for USSR for about 13 years. Notwithstanding the 1933 protocol, it is only after his being sentenced on January 5, 1940, to 8 years in prison for espionage and deported to the prison camp Norillag in Norilsk that the US authorities learned about his case.
US Secretary of State Cordell Hull, and Ambassador of the USA in Moscow Standley enquired about his fate. It is apparent in this telegram from Hull to Standley:
Please take up this case informally with the Soviet authorities and since Oggins is an American citizen request permission for an American Foreign Service Officer to visit him as provided for in the 1933 agreement, or that Oggins be allowed to appear at the Embassy.
Those efforts had some effects since on December 8, 1942, Oggins received visits from American diplomats at the Butyrka prison in Moscow. However, he wouldn't ever be released until his assassination in 1947.
Others
This is part a many efforts of the US Embassy to monitor the fate of US citizens (some of them, but not all, being communists) in detention in the USSR.
Earlier, Lovett Fort-Whiteman was another American communist, settled in Moscow and victim of the Stalinist purges in 1937. I found no trace of any US diplomatic activity to enquire about his fate. No hit either for Thomas Sgovio, who survived 16 years in the Gulag and later left Russia for the USA where he published his story in 1972.
You may also want to check how the US authorities treated the case of Noel Field, who was arrested and detained not in the USSR but in Czechoslovakia.