I say this as someone with peewee leavings down both sides of both cars where he sits on the side mirrors before attacking his reflection in the windows... Breeding season, male peewee, we named him Oliver; he's been visiting for several years now. He also attacks his reflection in all the windows around our house and our neighbours houses and eats bugs (he follows me around the backyard when I'm digging) and assists in driving off the parrots who munch on my fruit (in the right season; he doesn't help with the stonefruit).

If your neighbour doesn't want birds perching over his yard, he has the option of cutting back the vegetation to the fenceline (but not over it). Doing so will probably fix his issue without affecting the behaviour or environs of anyone but him. (Suggest you go take photos of the fenceline as evidence in case the person in question is a real fuckwit. Documentation never hurts.)

Magpies also eat bugs and garden pests. You feed them your snails and lawnbugs, right?