In reality you want both if you can. The currents you are talking about have to make it through both sides, not just one.
However, the ground plane is the more important of the two.
The ground plane is the reference voltage for ALL your signals. If a component on the right side of the board has a different reference from one on the left, they lose the ability to communicate with each other. As such it is very important to make that reference plane as "solid", and quiet, as you can.
The supply side, on the other hand, can be much less important. Devices, depending on the technology, can tolerate a significant difference in Vcc across the board and still communicate quite happily.
Further, the ground plane also provides a second function not related to carrying the power currents. The ground plane provides a great deal of EMI control and protection. It helps reduce the emissions from the board and provides a certain amount of shielding.
As such, if you only have the choice of ground or power for a plane layer, you chose ground every time.