There are rooms clearly marked with mask and droplet precautions on the outside of the door, and the staff will take a mask from a box next to the door, go inside to deal with that person, and then when they’re done they take their mask back off and go back to walking around treating patients with no mask. What do the people in those rooms have?

This is pretty standard, and maybe I can shed some light on it. You don the mask as you enter the room and take it off as you leave to avoid spreading the contaminate out of the room. The mask adds a barrier and reduces your risk of contracting whichever disease (and subsequently spreading it to other patients), and all the stuff it's blocked stays in or at the room when you shed it. So the people seen doing that are actually playing their part in keeping whatever that person has limited to the room they're staying in.

As for what people in those rooms have, it can be a lot of things, but it really is what it says on the tin. They have something that can spread by droplet, which ranges from the flu to stuff like whooping cough or, yes, COVID. The system to keep these things contained is pretty consistently updated and has worked well when implemented. We were all wearing masks everywhere for a time because COVID was spreading like wildfire, and concerns of people becoming contagious before showing symptoms with no way to reliably innoculate/vaccinate medical workers