Propagation across a surface
So there is a magnetic stud finder and an electric one. I will assume you are speaking of the electric kind. In these, there is a capacitive plate in the front end which is driven using two multivibrators. As a multivibrator is driven, the voltage on the plate drops to zero and then as it goes low, the plate charges back up. The rise time/slope determines the dielectric changes between your plate and the plate in the wall. This dielectric change is what we detect.

Since an the electromagnetic lines are disrupted by the secondary plate, this causes a change in flow of the field lines. Thinking of this in terms of your sensor, there is something blocking the field lines over the whole wall. This means that the thickness of the wall is proper in all areas. So your whole wall is a stud.

As for when the power is turned off, that's a great question. My thought would be that the wood is a bit wet. This damp wood is then carrying current. You did say it was bare, and it may be damp. Might want to look into waterproofing.