ZAB Center Controller reported an aircraft experiencing a pressurization failure descended from their assigned flight level without clearance.

Narrative

While working I heard a transmission that was carrier only (unreadable). I reached out in the blind; saying that it was unreadable and to please repeat. Right after; Aircraft X started descending from FL390 without clearance. This was an immediate loss of separation; as Aircraft Y was ~3mi to the West of their position. Thankfully both aircraft were tail-to-tail; avoiding a potential head-on situation. I reached out again; and asked Aircraft X to verify altitude with no response. I instructed Aircraft X to maintain FL390 with no response. At this moment I knew the aircraft was descending to an unknown altitude and looked ahead for potential head-on traffic. I called Sectors XX and XY to coordinate the unknown descent; and instructed Sector XY to turn one of their aircraft at FL360 to prevent an imminent situation. I then returned to my frequency and asked Aircraft X to ident if they were [requesting priority handling]; in which they did ident. I told them all of their transmissions were unreadable; and that if they were wearing oxygen masks that may be why. They responded that they needed a rapid descent to FL210 for depressurization. They started turning off course; in which I asked them to ident if they were planning on diverting; which they did not. Aircraft X then said they needed to descend to FL180; which my D-Side coordinated with Sector XY. After issuing the descent to FL180 I made sure the pilot was OK with a frequency change (as they were ~20mi inside Sector XY's airspace at this point) and they accepted the frequency change.Not much we can change in a situation like this; it would've been helpful to have audible transmissions so that I could more efficiently move aircraft out of the way - but this is unrealistic in an emergency situation.