All Droyne have it. Only Droyne trained in it can control it. It becomes an involuntary reaction without training.
Speaking on do only this race or that race have such and such a power:
True, you can offer the traditional powers that are specific to one race to characters outside of that race if you like, but then you dilute the coolness and uniqueness of that race.
Personally, if I had a player who really really wanted say "Zhodani" as a psionic skill choice, but their character was a 3I Vilani-Solomani mix, I would not let them do so on a whim. They would have to create a very interesting character narrative as to why this was the case that fit their character's event history timeline as rolled. Then *maybe* I would let them be that rare aberration of a non-Zhodani trained in Zhodani psionics. Same with Droyne powers as well.