Dispatcher reported being notified by flight crew of a loss of pressurization. The flight crew diverted to the alternate airport and landed.

Narrative

Crew sent an ACARS stating Loss of Pressurization and asked how ZZZ [airport] looked. I noticed they were already descending to 10;000; and looked over ZZZ to make sure it was suitable. I sent them a message and said it could work. After that I coordinated with ZZZ Operations and they said they could not handle us; but I told them this was a [priority situation]. I asked a fellow Dispatcher to contact the Maintenance Dispatcher to inform them of the situation. I called the manager and told them the situation; and they got ahold of Operations to work out a gameplan with them. Meanwhile I was still talking with my crew and asked if they ]requested priority handling] and if the masks dropped. They did [request priority handling]; and the masks did not drop. I sent them the landing data and made sure we weren't landing overweight. Once the crew was on the ground I was able to go over what happened with them; and they told me there was a differential pressure increase above 8;000. At first; they noticed their ears popping and that's when they took a look at the instruments. The cabin pressure began decreasing as they descended; but the masks never deployed. The supply fan was flashing off as well. Before informing Dispatch of the situation; they ran the checklist and determined to descend to 10;000. I trusted the crew's judgement and given the information available to me at the time event I felt it was safest to head into ZZZ instead of overflying it.