Where I saw this, households would generally pay for a 2hr guest permit (maximum one per household, or maybe two) that they lend visitors to display in their windshield, and if someone needs to stay longer, then you pay for a temporary two-day guest permit at the parking office. Either way, the permit is used for street parking. Finding a spot can be a minor adventure but is usually doable. You may have to walk a block. If someone is visiting for longer then they probably need to pay for a spot in a parking garage, or just -- fly in and leave the car at home.
So then the city has to issue parking passes plus guest passes, and then actually enforce them. Seems totally impractical for all but the densest cities.
Sure there's some administration but it can be mostly handled digitally nowadays. I've seen it work in many cities that are not particularly dense by European standards.
The enforcement are handled by the same people that do normal parking enforcement. They scan the plates, see if there's a valid permit/ticket and write a ticket otherwise.
Which "people" do normal parking enforcement? Most cities do zero parking enforcement in residential neighborhoods unless someone specifically complains.
This is indeed one of the biggest problems with such parking permit scheme.
What happens is the resident needs to move their car to the street (possibly driving around for a long while to find a spot) so the guest can park on their driveway. It's a pain for everyone involved. And you better never have a party where more than a couple people visit at once.