Electronic – Does the drive circuit for a piezo buzzer require extra power

The battery in question would limit current to the buzzer, per the battery's internal resistance. Thus, the buzzer "demanding" more current is not the constraint, it would only get as much current as the battery is able to supply.

Typical piezo behavior with limited current available (and I just tried this to check) is that the volume of the buzzer starts dropping sharply once the available current reduces below around 30-40% of the "rated" value, but it remains audible (faintly) until well under 5% of rated current.

Also, a piezo buzzer's current demand consists of sharp pulses, much like a capacitor being charged and discharged by a square wave. Thus, the maximum pulse current rating of the battery (10 mA) will come into play, rather than the continuous current rating.

In short:

The buzzer will work fine, it just may not be as loud as it can be. Given that even a tiny piezo buzzer (mine has a minuscule 9 mm diameter) makes an awfully loud sound, this may actually be a good thing.

You said you were a novice, so here is something simple:

The page you linked to turned out not to be a datasheet, so we don't know anything more about the piezo speaker than what you say. The above circuit should work well enough for most piezo speakers, although it can possibly be driven from a higher voltage, which would yield higher volume.

R1 limits the current thru the base of Q1 when the digital output is high. Figure the B-E junction drops 700 mV, so that leaves 4.3 V accross R1, which causes 4.3 mA base current and will draw that much from the digital output. Most digital outputs can source that much without issue. Check your datasheet. If not, make R1 higher. You most likely don't need as much base drive as this.

When the digital output is high, Q1 will have 4.3 mA of base current, which should easily give it enough current sink capability to pull the collector as low as it can go. Figure you can count on a gain of 50, so that means 215 mA of collector current, which should be way more than required.

D1 is there to give the kickback current of SP1 a place to go when Q1 turns off suddenly. Piezo elements have a significant inductive component as viewed from the driving circuit. If you don't give the kickback current a place to go, then the voltage can get high enough to fry Q1. R2 cause some back voltage to this current to make it die down more quickly, helping the high frequecy response. R2 can probably be a lot lower, but again, you didn't supply a datasheet so all the component values are just rough ideas or guesses at best.