Yeah, I will at least try to speak with the neighbor about it before I do anything. With the way the properties are set up, they barely even see the fence, whereas for me, it's kind of front and center in my backyard. I expect they may be surprised when I point out that it's technically their fence.
It is a chain link fence. A top rail sounds like a good idea which wouldn't be too unattractive or hard to install. I'd sure love to replace it with a proper wooden fence because the privacy in my backyard is severely lacking when they're on their front stoop or in their driveway, but with the grade and the need to remove the current fence, it would be very expensive.