Some people, although listening to another's kidush (Friday night), sing part of it along with him: specifically (in my experience), the lines starting "ki vanu vacharta" when recited with this tune. This seems to me arguably not to be a problem (see e.g. Mishna B'rura 690 s'if katan 13–14; but see the comments on this question). But I'm wondering whether there are any sources that discuss the issue.
As always, for practical halacha, consult your rabbi.
To attempt a different parallel, the Mishnah B'rurah in Siman 124, S"K 16 says that during the repitition of Sh'moneh Esrei, people should not sing along with the Sh'liach Tzibbur because it is k'yuhara and like kalus rosh.
I had always understood that one should not speak/sing along with the one reciting kiddush because of the principle of Trei Kali Lo Mishtamai” -- two voices cannot be heard simultaneously (see Megilla 21b).
The Star-K says this as well here when listing the conditions to fulfill one's obligation to hear kiddush
One must hear the entire Kiddush – One should not speak while listening and should not say “Baruch Hu U’Varuch Sh’mo.”
If one spoke, b’dieved, the following halachos apply: If while speaking one failed to hear a word that is integral to Kiddush (e.g. “Boruch” or “Hashem Elokeinu” after “Boruch Atoh”), one is not yotzai Kiddush. If one did not hear a word that is not integral (e.g. “Atoh” or “Kee hu yom”), one need not repeat Kiddush. [...] One should also not speak between Kiddush and drinking.
