capacitive proximity sensor - how does it work
No, the bed is not the second plate.

Those sensors contain both plates of the capacitor, but they are arranged so that the area in front of the sensor is part of the dielectric volume.

If some material is brought into that area, its dielectric constant  and its distance to the sensor changes the capacitance. This allows to detect almost any type of material (your idea would work with conductors only), though calibration is needed to adjust the sensor to the actual material. 
When the object has a large, flat surface, the sensor can be used to measure the exact distance as well.