I learned the song at a Jewish summer day camp in Los Angeles, CA, in about 1962. A male counselor taught it to us and it was probably THE most popular song of the ones we often sang that summer. The lyrics were somewhat different than the ones elsewhere in this thread, but it's the same song:

Can you guess (3x) who we are
We come from afar from the land of the car
Can you think (3x) who we can be
We're Lenin-Trots, the Bolshevaks, we come from across the sea.
I'm Lenin, he's Trotsky, the Russian diplomatskis
We have overthrown the czar, now we are the conquerors,
Hi-ho (3x)
Weeeeee're the Russian diplomats!
Hey, Russia-Pisha, Pussia-Pisha, hey, pell mell; hi-ho the derrio, the farmer in the dell.

(The one additional verse was the "landed here at Ellis Isle" verse.)

Thinking back on it as an adult, it has, as others have written here, a vaudeville or musical theatre feel to it...maybe from a revue in the '20s.