Having the output as headphones how are the passive pickups powered in an effects processor ? Do they take up the power source of the processor itself ?
The pickups are "powered" by the movement of the strings. The effects processor amplifies the signal until it's powerful enough to be heard. If you plugged your headphones in directly, there wouldn't be enough signal even when you bang hard on the strings to hear anything through your headphones.
A low level signal is generated by the vibrations of the strings above magnets. This signal is then boosted by a preamp at some stage in the process, whether it is an effects pedal (in your case it sounds like it is), or a power amplifier.
Active pickups differ in that they have a built in, powered preamp which gives a higher level signal before it gets to an effects chain or amplifier. If your guitar has a battery in it, it almost certainly has at least one active pickup on board.