You will really have to talk to the folks next door about your plan for a fence. You can't just put another one up without having some kind of discussion. If you go ahead and do it, and don't say anything to your neighbors, that will create an unpleasant atmosphere, which won't go away. Next time you see each other outside, take that opportunity to say something like, "Do you have a couple of minutes to spare?" That should be an easy intro to a fence discussion. You don't have to use legal terms, or anything remotely official. Just say what you plan to do, and then ask them, "What do you think?"