There are various companies selling insurance by the hour. I'm only aware of see their ads in the bogs at Motorway services but no doubt they publicise themselves elsewhere.

The “other cars” provision of course only allows you to drive cars that are also insured in their own right. As long as the car you are selling is insured and the buyer has “other car” cover they can test drive.

Otherwise, as you say they need to arrange their own insurance. There are online short cover services that offer instant cover.

The problem remains that you are still at risk if for example they lie about their claims record rendering the cover invalid. Or if they turn up with apparently good “other car” paperwork but they have cancelled the policy, or faked it with a printer. Maybe I’ve seen too many Police Camera Action type progs where even the Police have to dig to verify apparently valid cover but I would not let a stranger test drive my car.

The only fully safe option is to do a demo drive or trade it in to a reputable dealer or buying service.

How do private car sellers deal with situations such as this?

It's quite simple. If the buyer seems serious I take the car for a 10-15-minute drive with him/her as passenger. It's never wise to hand the keys to a stranger, especially if the car is worth a good bit (mine usually aren't). If the buyer insists on driving, demur politely - until they hand over the cash.

Sell to a car dealer! That's one of the many disadvantages of selling cars privately, I'm afraid.

There are actually several different insurance companies that will sell you just one day's car insurance. If you had someone coming to view a car which you are selling then you would have to ask them to do this and show you proof that it had been done, of course.

Or you could take out the one day's insurance yourself, should you feel so inclined.