It's an interesting question but not one I have a lot of knowledge about. I'd imagine that the most common form of illness found in hostels would be from food poisoning, especially in countries where this is a particular problem. There's not much you can do about that but wait it out, but it certainly isn't pleasant for the person or those around them. I guess a vomiting bug such as Norovirus could also spread round a hostel - again all you can do is wait it out. I'd certainly try and give the person a private room if it were at all possible.
When I used to work at a large hostel chain we were told we were never to give any first aid ourselves, or even give out simple pain-killers, due to the risk of litigation from someone claiming that our help had in fact hurt them. Personally I think this is ridiculous - surely if someone was in obvious need of help you'd help them? I remember one guy who'd cut his head on the lid of the trunk of his car and there was a lot of blood. Obviously we provided bandages and drove him to a doctor - you're hardly going to leave the guy to bleed to death.