Why are power supply decoupling capacitors used on both supply rails in dual supply opamps?
The supply current for both supplies are different most of the time. Only if the output of the opamp with dual supply is at ground potential and the output current is zero both supply currents are (almost) equal. If the ouput jumps positive, there will be a short increase of the positive supply current and a decoupling capacitor is needed for positive supply. If the output jumps negative, the other decoupling capacitor is needed. If the output voltage of the opamp changes dynamically, both decoupling condensators are necessary. If there are no decoupling condensators, the opamp may oscillate in some cases. Decoupling capacitors are especially important for fast opamps. Both capacitors should have short connections to the supply terminals of the opamp and to ground.