Controller reported an aircraft descended without permission or communication with ATC twice; due to a pressurization problem resulting in a loss of separation.

Narrative

Aircraft X check on frequency at FL390 requesting FL350. I told him I have his request. I called for control. I couldn't descend to FL350 yet because of traffic at FL370 within 3-4 miles. Aircraft X said he needs to descend now so I turn him; assign him FL380; and call the ZZZ1 traffic for him. He tells me he's looking for the traffic and reads back descending to FL380. I see his data block update to FL377 and ask him about his descent. He tells me he can't maintain the altitude and is going to FL350. I turn him again but lose separation with the other traffic. I had about 4.5 miles going through the altitude of FL370. I ask what the problem is; and he says he's having a pressurization issue but that FL350 should be fine for now. But soon after requests to go lower and eventually we [gave him priority handling]. Just better communication. He said he would like to descend now but did not tell me the reason. If I knew there was an issue I could have given more urgency to vector and descend. I understand it's a possible situation and a pilot is worried about flying his aircraft first. Maybe I could have prevented the separation loss.