It doesn't work quite the same way in Greek and Aramaic. In Greek, they're halieis (fishers), and Jesus says they'll be halieis anthropon (fishers of men). Incidentally, the KJV translates it the same way--they're "fishers" who become "fishers of men".

The Syriac (an early translation of the NT into a dialect of Aramaic) does the same thing, with calling them ṣyd' "fishers" who can become ṣyd' dbny 'nš' (fishers of humans). However, that word for "fisher" isn't attested in Jewish Aramaic (it means "hunter" or "trapper"), using the root dg instead. But the pun would still exist.

tl;dr the fisher/fishing people thing exists, but the "fishermen/fisher-of-men" thing is just English. It requires compounding -man, which doesn't exist in other languages.