Incorrect. Gaúcho here, born and raised - but with many many friends and relatives in the other mate-drinking countries and across our (unfortunately currently under a lot of distress) state.

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.