In most Nusach Ashkenaz shuls that I have attended, the congregation and chazzan sing Lecha Dodi and Kel Adon together.
In most Hassidic minyanim (they are Nusach Sefard. I say "Hassidic" since the majority of Nusach Seafrd shuls I have attended are Hassidic shtieblach) they do not sing together, but, rather, the chazzan sings a verse, then the congregation repeats it (a.k.a. sung responsively, which is specifically not the topic of To sing or not to sing - Piyutim on Shabbath).
How did these different minhagim originate?
Also, is there a halachically preferred method?
(I think that this is a general rule for all cong. "songs" but these are the only two that I can think of where I see differences.)