This is not quite right. As of patch 3.13 at least. The SoZ doubles the War Weariness, not the Accumulation.
If you attack a Civ that has SoZ and you have 800 War wearines Against them, it means you accumulated 400 WW by combat actions, which is doubled by the SoZ to 800. The Moment you take the SoZ city, your WW against the Civ drops Back to 400. You loose the city again, and you are back at 800. (Assuming taking/loosing did not cause extra WW for the moment)
I do not know what happens to his WW, as there is no way to look it up, but i assume it works by the same rules. However, if you are attacking 'he' will not have much WW anyway, and also the AI gets discount on WW anyway.