A buyer intending to buy a car privately, and to test drive the vehicle themselves will need to ensure they have adequate insurance cover. As the car is not yet theirs, it is not possible in most circumstances to get fully comprehensive cover, as there is no insurable interest in the vehicle.
But the seller will need to ensure that the potential buyer is covered for both third party cover (otherwise the seller, assuming he is the registered keeper, will be open to a charge of driving, permitting or causing the use of a vehicle without insurance) and it would be beneficial to have fully comprehensive cover for the potential buyer aswell.
It is not unheard of for potential buyers to crash a seller’s vehicle, leaving no traceable address and other necessary details. So be very careful when selling a car! It is very tempting to just allow that “quick” test drive, but it could go so horribly wrong!