I think in every shul or yeshiva I have ever davened in on the Yamim Noraim or aseres ymei teshuvah, when the shat"z got to the spot where you insert zachreinu l'chaim, mi kamocha, ksov l'chaim tovim, etc., the shat"z would pause and everyone would say those lines out loud together, and then the shat"z would say it and continue.
The GR"A in Maaseh Rav #204 writes that this is incorrect. According to GR"A, the shat"z should daven straight through with no pause and no interruption by the tzibur. Like most everything else in Maaseh Rav, there is no explanation as to why the GR"A held this way, but some of the meforshim link it to the GR"A's opposition to the tzibur saying "baruch HU u'baruch Shmo" during chazaras ha'shat"z (Maaseh Rav #43) and suggest that the GR"A treats chazaras ha'shatz as if the shat"z is being motzi the tzibur in the brachos of shmoneh esrei, so no interruptions are allowed.
I'm not sure that works, because 1) the GR"A does allow piyutim to be inserted, 2) and even if you treat chazaras ha'shatz like something you have to yotzei in, the insertions of zachreinu etc. are part of the shmoneh esrei recited by the shat"z, unlike baruch Hu ub"S. Even if the tzibur is being yotzei, why can't the they say a section of the bracha themselves instead of relying on shomeh k'oneh? (I believe whether you can reply on shomeh k'oneh for only part of a bracha is a debate in Achronim, but it would be a chiddush to say the GR"A is taking a stand on that issue here.)
There is a third minhag that is interesting. Some are noheig to only recite the last two insertions, the ksov l'chaim tovim etc. out loud during chazaras hashat"z. The logic here is that the insertions build upon each other. First we ask zachreinu l'chaim, then we build on that and ask not just l'chaim, but ksov l'chaim tovim. Once you ask for that in your shmoneh esrei, to go back during chazaras ha'shatz and just ask zachreinu l'chaim again without tovim is a step down, a step backwards. That we don't want to do.