In the film Captain America: The Winter Soldier, a flashback explains how
Bucky was 'recovered' after his fall from the train.
I can't recall whether it is explicitly stated that he fell into the custody of the Soviet Union, but I know this to be the case in the 616 and he speaks Russian in the film.
The bunker under Camp Lehigh contains
Arnim Zola, who provides a piece of expository dialogue describing how he was acquired through "Operation Paperclip", colloquially known in Europe as The Brain Drain.
At the conclusion of WWII, Europe was divided up between what would eventually become the EU and the USSR. In order to keep players off the table, a great number of European 'assets' were repatriated as Americans:
Arnim Zola being one of them, and it is implied that the Winter Soldier is another.
If the Winter Soldier was already a Soviet asset, how have HYDRA (a German/American brain-drain coalition of Fascism) come to be in possession of him? If the Soviets created the Winter Soldier as a weapon against American-Intelligence/HYDRA/SHIELD, why is he fighting on their side? does the film explain this at all, or have I missed something?