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?
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.