The recording I grew up with, in my parent's record collection, was
the Elektra Young Man and a Maid album
with Cynthia Gooding and Theodore Bikel.

"Proshchay," the farewell song, is fittingly
the last song on the album.

So the version I grew up with is the duet version.
Not a word of English, unlike
the Limeliters transcription presented in this thread
-- the Limeliters recording is unknown to me.

The all-Russian word transcriptions I see on this thread
nicely match what Gooding and Bikel sing together.
Bikel sings the first verse;
Gooding and Bikel follow with the chorus, in harmony.
Gooding sings the second verse;
Gooding and Bikel sing together on the following chorus.
Bikel sings the third and final verse alone;
and the two of them close out the song in duet.

The above all sounds very formal and decorous.
Actually the recording is nearly uproarious.
It comes across like a comedy routine.
Cynthia Gooding sings the Tin Pan Alley Russian,
if that's what this is,
and kind of hams it up, cabaret style, with much emotion.
But Theodore Bikel is diabolically humorous,
messing about with the Russian syllables rather like
Tevye singing If I Were A Rich Man with the
ya-ba-di-ba-di-ba ya-ba-di-ba-di-bas, well not exactly that
but it sounds very similar.
The only accompaniment is one guitar,
I am guessing that Bikel plays it.
And as there are no other backup musicians,
the two singers can mess about with the song
without distracting anybody else.

At the very end, the tempo gets cranked up faster,
and Gooding throws out the lyrics altogether
and starts singing the chorus to a full-throated
ay-yay-Ya yay-Ya yay-Ya! while
Bikel ad-libs in harmony.   It's a hoot.