Do residential smoke alarms monitor their sounders? If so, how?
How do you propose that the smoke detector tell you its buzzer is broken?  If the buzzer is broken, then it can't buzz to get your attention to notice that it is broken.
There's really not much that can go bad on a buzzer.  They are electrically and mechanically pretty simple.  They are inside the housing, so physical damage from outside is unlikely - you'd see the broken detector housing.
I suppose a bug could get in through holes and jam up the works, but at worst the buzzer would just not be as loud as it ought to be.
The residential smoke detectors I've looked inside of (from my own house when installing new ones) only had one buzzer.
Three pin buzzers use the third pin as part of the oscillator circuit.  I suppose you could detect the oscillation that way, but you have no way to signal a failure.
Periodic test of the smoke detectors is your best bet.  Go around and push all the test buttons and make sure they all buzz.

Smoke detector systems in commercial areas are different.  The individual detectors may (or may not have buzzers,) but the system as a whole will have multiple sirens in various locations that will be triggered in groups depending on which detector triggered.
There's redundancy built in in that regard.  No zone will depend on just one siren or just one detector.