It depends! Generalizations about taxis are usually false. In San Francisco, cab drivers do own medallions -- you actually have to drive a cab at least 800 hours per year in order to own a medallion.
Not every driver in SF owns a medallion. Drivers who are owners can and do rent out the medallions to other drivers for when the owner is not himself out driving. In this way, medallions can theoretically be utilized 100% of the time. The way it works in SF is that most drivers are affiliated with a fleet (they are not employees of the fleet). The fleets typically (though not always) own the vehicles. The fleets rent the medallions and then hand them out along with the vehicles to the drivers, and the fleets also handle insurance and other infrastructure stuff, plus dispatch. The drivers pay a "gate fee" at the start of their shift to the fleet, and then keep their earnings that they make during the shift (so they start out in the hole and earn out their gate fee (hopefully!) and then anything on top of that is profit to them).
It may be completely different in other cities. My understanding is that in NYC, drivers are typically not medallion owners, and there aren't really fleets in the SF sense -- instead there are taxi yards that offer some of the same functions, but have a less exclusive relationship with taxis and some kind of different business model. But I don't know much about NYC. In cities with less expensive medallions, I'm sure it works differently again.
Source: I am an employee of Flywheel -- we work closely with SF cabs.
EDIT: By the way, again in SF at least, where medallions recently cost $250k (they've gone down since), you might ask "How does a cab driver pay $250k"? The answer was simple: they get loans. There are or were bank loans available to get a medallion, on the basis that medallions gave you the earning power necessary to pay back the loans. So, given that banks are not charitable organizations, if you're wondering who got the excess value created by the legal monopoly of the medallion, part of the answer is "banks."