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 impossible to say from this photo. Do you know what species it is? A closeup of the leaves would be helpful, and a picture of the entire tree.
Either way, best option is going to be to get a certified arborist out to look at it. It's likely in need of a regular ol' prune given those dead lower branches in the pic, but an arborist could tell you if there's anything more major to consider. An arborist can assess the tree from your property and trim branches over your property, but any major work will require consent of the owner of the property the tree is on.
My recommendation would be to get an arborist assessment and then approach your neighbor with that in hand and talk through options they're willing to consider. Cutting down a tree or even a big part of a tree this size should be a last resort, though
Thanks, yea I think im going to get an arborist
Here's about as good of a shot of the whole tree I can get
Those dead limbs under the leaves definitely need to go. Not sure about the whole tree. In my experience an arborist is more aggressive about keeping trees and will recommend treatments that aren't always worth it.