No, current in the first wire does not induce current in the second. It's the derivative of the current that induces a voltage (open circuit) or current (shorted) in the second.
In the cases where there is power transfer (the second wire not open, for example), you can detect the loss of this power in driving the first wire.
What you are describing is a transformer, just that it has a unusual geometry and the coupling isn't very good. Just like with any transformer, power coming out the secondary came from power put into the primary. With the right circuit, the power being put into the primary can be measured, and thereby varying power being drawn from the secondary can be measured.
Keep in mind that when the secondary is open circuit, it's basically not there from the view of the primary. The transformer then degenerates to just a inductor.