Why does my stud finder report an AC warning over the entire wall?

The stud finder and the volt stick operate on changing electrical field strength levels, and they perform measurements by means of capacitive coupling.

It is possible that the fire & sound insulation in the parting wall is statically charged due to your or the neighbour's electrical wiring through or against the insulation. Even with circuits disconnected at the breaker, there can still be a small enough field capacitively coupled-in to trigger your devices.

With a bit of luck you may be able to reduce the sensitivity of your detector by placing a non-metallic spacer, like several sheets of paper, between the device and the wall, as you skim the wall.

Have you asked the neighbour to disconnect their power?

Alternatively, if you have outlets or wiring in that wall, you might be able to neutralize the field by grounding all conductors in the cables. This can be done by connecting a load but with the circuit power off. The load must preferably be "dumb" or "old fashioned" without control circuits, so no LED light, but a simple incandescent bulb, fan, or kettle would work.

Another trick that might work is grounding the wall itself: while grounding yourself place your hand on the wall to dissipate and keep dissipating any coupled charges. The important thing is to make sure you are well grounded, and so it depends on the flooring and your footwear.

It may be safer and more effective if you grounded the wall by placing a large and bare grounded metal object against it (e.g. pressing/leaning some HVAC sheet metal), or smaller objects at several spots on the wall. Then you have free hands and a clear path to ground without placing yourself in the circuit path. You could also ground the insulation by piercing nails or screws at several locations into it, and grounding them.