Howard, the electric fence is obviously not just a tool for you to keep animals in and out, it's also a hobby. LOL
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Who's privacy fence is it? Yours or the neighbors?
Either way, unless you are going through some type of blood feud with the neighbor, I'd run it on the outside, and just below the top of the fence. That way, any climbing predator is going to have to get past it to get over the obstacle. A two wire obstacle. One of them hanging on this insulator, with a 6 inch hold out?
https://www.amazon.com/Red-Snapr-Extender-Insulator-IW5XNYRS/dp/B007PARUDI/ref=pd_day0_86_2?ie=UTF8&psc=1&refRID=WPASYXQ1R7P6NBWVJCSW
The other on a similar insulator, but not held out so far. More like 2 inches and just below the held out wire. A raccoon or such will likely try to slip between the wires......one hot and one grounded.....and should get zapped in the process. I'd be inclined to start with the hot wire being about 2 inches BELOW and on the short insulator.......so when the coon tried to slip between them, likely his foot will still be on the hot wire......the top outside wire being grounded and it I'd think it would send him flying. With a ground wire up high on the fence, it could be wired direct to the fencer to assure a good ground.
I would explain things to the neighbor and let them know what is going on. It should be at least 6' high, so well up off the ground within reach of any kids or pets.
I would not want it on top.......birds could land on it and not entirely certain varmiit will touch it AND the ground there anyway. If forced into it, that would be my 2nd choice, but still retain the 2nd ground wire. You are correct, the wood fence is not a reliable ground.
Second issue with the wood privacy fence is how to keep a predator from digging under it if it is on a property line. The apron or hot wire needs to be on the outside. If you used a hot wire on your side, it would need to be nearly at ground level so when they surface on your side of the fence, the first thing then encounter is the hot wire. Not entirely foolproof.