Detached home - so I am presuming you mean next-door neighbors, not renters upstairs or downstairs.
I have heard of complaints from the fan nosie from high-efficiency furnaces - but not of outside noises getting into the system and being disturbing throughout the house before, but I guess if the vent pipe terminates pointing straight at their house I can see it happening.
I would check the installation manual - most can have either a Tee put on the end (so the exhaust goes both up and down but no longer a direct opening aimed at the neighbors) - many you can also put a normal direct-vent woodstove type vented end cap on which would block the neighbor noise somewhat, though not as much because metal and the metal end would be facing them still.
Or you could go with the screening thing (located several feet from the exhaust outlet) - vegetation, wood lattice fencing, etc between you and the neighbors.
Water running - assuming no outdoor plumbing changes when the water heater/hot water tank was put in (unless you are in a real warm area so it is located outside the house), I can't see that making any difference - unless it also is very high efficiency to also open to their side at the exhaust, where I can see maybe some running hot water sound might make it through the exhaust to them.
All you can do is walk around the house while water is running hear where the sound is appearing outdoors - and when they are talking etc listen at your furnace exhaust and ducts to see if that sound is coming in through the exhaust pipe or not and modify if needed. note that nothing should block the exhaust flow - manual will show required free space around it - typically 3-5 feet.
Device detecting every movement in your house - if you mean a high-amplification acoustic parabolic listening device or such, if you actually catch them using it on you you can call the police on them under invasion of privacy or "peeping tom" or unlawful recording or similar laws - but the police would have to catch them doing it. Though a surreptitious cellphone photo of it or two in use might help convince the police they are really invading your privacy.
