The tree is an adjoining neighbors backyard with large branches that come pretty close to my house (Though there is a sizable gap, if it comes down under the wrong wind direction...) and are actually touching another neighbors roof.
If it is unhealthy, is it recoverable? I'd rather not have them cut the whole thing down (it provides a lot of shade for my backyard (well, when it's healthy that is lmao)) Should I talk with them directly (I don't really know them, at all) or let the city know?
It's hard to tell from the pic cause I can't get a good view of the leaves, but based on the bark, it looks like maybe it's in the willow family. I'm thinking populus deltoides (cottonwood), but I could be way off. They often have a lot of dead branches, especially in the rocky mountains. It seems like your neighbors tree might have broken a little bit in that crotch, allowing infection to get in. It seems like there is a bit of slime flux causing that staining. If it's close to housing, it is worth having an arborist look at it.
Like someone said below, do some due diligence to make sure you are getting a certified arborist cause otherwise, you'll probably just find a "tree guy" who will automatically just say it needs to be trimmed or taken down. There's a decent chance the two sides of the tree could be bolted together to reduce stress on that joint.