What does 0 and 1 mean in logic gate (NOT gate to be specific here)
Usually such gates are voltage-controlled devices, and not current controlled devices. They generally have very high input impedance, which means that very little current is drawn from the input. How much, depends on the device, so see the datasheet for that.

The output side is usually meant to be likewise voltage indicator of the logic state. Whether the is any current flowing out depends a) the state of the logic gate (current can be drawn only from non-zero voltage) and b) on the input impedance of whatever is hooked up to the gate: if the device connected to the gate output has low impedance then much current will flow from the output of the gate. How much current such a gate can source also depends on the specific gate.