One looks into their Siddur on Rosh HaShanah and will find during chazaras hashatz many things for the kahal to be saying as well (either in a response form or together in the chazzon, etc.) It’s not clear to me though what does one “have” to say and what is just minhag, etc. I realize there are many piyutim that many kehillas say however I’m not so much asking about that since it’ll vary between machzorim and minhagim of kehillas, etc. My main question is when davening on Rosh HaShanah, especially during Chazaras HaShatz what is required to be said? Is it only Kedusha and Modim or perhaps more things that are “requirements” in the tefillah of Rosh HaShanah (while I’m sure it’s nice to say everything I’m asking from a purely halachic aspect.)
There are no formal obligations beyond kedushah, modim, and answering amen at the right times.
The piyutim are a matter of custom, and this depends on the practice of the community you daven with.
See Maharam Mintz (87), who writes that because there is no obligation to say the piyutim, they should not be said by a yachid, because of the problem of making a hefsek (he discusses the piyutim of birchos shema).
Another sefer I saw writes that one should say piyutim only if one has kavanah, because they are not obligatory.