Note the double bars. Most times the Gloria
is sung in alternance. The Priest will intone “Gloria in excelsis Deo”, then the cantor or schola will sing “Et in terra pax hominibus…” up to the first double bar, then the assembly or choir up to the second, etc.
As one who sings in a schola, and along with the monks at the abbey, it would cause problems to sing the parts one isn’t supposed to because it throws the schola off their rhythm. It is they who set the rhythm and pace by the parts they normally sing. Monks hate it when they’re thrown off (well, at least the perfectionists of the Solesmes congregation

I’ve been to some abbeys where it doesn’t matter so much). I chant it from both POV; in our schola I sing the schola parts along with other members, and we lead the laity into singing the other parts. At the abbey, it’s the other way around, I respond with the laity.
Whether it’s rude or not… well I’d be attentive to the surroundings. If the mood is very informal and everybody just sings along regardless of ability, then by all means. But if the mood is very formal as at our abbey, then discretion is the better part of valour…