Correct, that’s a thing only in Rio Grande do Sul.
Source: born and raised there :)
Source: born and raised there :)
Everywhere in the Pampas region - Uruguay, Paraguay, Argentina and Rio Grande do Sul -, the "roda de chimarrão" is pretty much the same. (Apparently in some part of Uruguay it's a bit less shared, more personal).
One person holds the thermos and generally owns the mate. They usually have the first drink as both a way of validating the drink and as a niceness to the rest of the group, since it's the coldest. Gourd is refilled and passed on to the next person, who drinks all of it and passes it back to the "cevador", who again refills it and passes on to the next person. So on and so forth.
I grew up in a border town bordering Uruguay and over there I’d definitely see folks carrying their personal thermos and cuia. But I would see groups on the Brazilian side of the border.
Uruguayans tend to be more likely to have one thermo and mate per person, similar to the tradition in Corrientes (which makes sense if you look at who their immediate neighbors on their western border are).