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?
In don’t think anyone can tell you from a photo. You need an arborist to look at it.
You can talk to the neighbor and express your concern. Ideally, have a paper trail in case something does happen.
You can have an arborist trim any branches hanging over your property without asking the neighbor.
I would get an estimate from a couple arborists for the trimming what’s over your yard. Then, approach the neighbor and offer to pay that amount towards having the whole thing trimmed—must include trimming the parts over your yard.
I wouldn’t pay until the work is done.
If you don’t know the neighbor at all and/or can’t be sure they’re trustworthy, I’d just get the part over your yard done.