The nice thing about having ground and supply planes on opposite sides of the board, other than the ease of routing, is that they form a nice big decoupling cap that smooths out your power supply.
No, its a fairly small capacitor, order of 1nF for an Arduino-sized board - the important feature is the low-inductance path to the real decoupling caps. A standard 100nF or 220nF decoupler far exceeds the board capacitance!
The basic rule for decoupling is having a low-inductance path from the supply pins to the decoupling caps - lets imagine switching 100mA in 10ns - that's 10MA/s which is going to cause 0.3 volts across 30nH - roughly the inductance of one inch of PCB trace... If the decoupling cap is that far from the chip and with only normal traces, its supply voltage will fall or rise 0.3V during the switching event depending on the sense.
A plane has a much lower inductance (wider traces are better, a plane being the ultimate). So long as each chip has a decoupler right next to it one plane is plenty - having two planes means that the siteing of decouplers is less crucial and one cap can serve many chips.
In practice the planes are rather broken up on a 2 layer board so it may be better to have ground plane on both layers to provide a single plane with better continuity - the continuity also means that long signal lines always have an adjacent ground return path which keeps their inductance down too. With 4 or 6 layers its simple to have continuous plane(s).