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Hello, I've searched everywhere but I can't find a simple explanation of the multiplayer mode. Does it allow 2 people to work together running the same hospital? Or is it simply some kind of score/stat comparison at the end of each month?

I think you run separate hospitals and you can send patients to each other's hospitals (just like you can send a patient to a competing hospital through single-player). The netcode is incredibly dated, so I've never actually played it.
You play against each other in the same town basically competing for the best stats, although there are a few game modes like the 1st person to fail an emergency loses or 1st person to fail a VIP visit loses, there are some others too. You start off with a small building each at opposite ends of the map then when you need more space you each bid for new plots. If you want to sabotage your opponent's hospital you can drop a Litter Bomb in their hallways which explodes after a minute or so throwing crap all over the place and causing rat infestations and whatnot :)
It sounds like you're playing largely separately then? It seems that it is a very limited multiplayer version of the game. Would anyone recommend it? Is this something I could play with my partner? Thanks