This reminds me of a text conversation I had with Mosh last night about whether the Dor can be channeled on Roshar, and if so, how. Keep in mind, we have seen disguised Elantrians on Roshar, but we haven't seen them do any magic, so they could have easily put on disguises before leaving.
But did they? And by the same token, does everyone in the Cosmere have Breaths (in the same sense that people on Nalthis do, that is), or did Endowment's influence on Nalthis turn those specific pieces of people's souls into something that can be manipulated, making Nalthis-dwellers the only people in the Cosmere who have, in a sense, "tangible" Breaths?
We need more information. Right now, I'm going to guess that Awakeners simply bring Breaths with them, and Elantrians worked their disguises before going to Roshar, because based on what we've seen of the differences between magic systems, that seems to make the most sense in my mind. That's just a guess, though, and can easily change once Brandon reveals more to us.