If at any point during the car buying process unsure of a car's history or the validity of what the seller tells you then its a very good idea to get the car history checked out. This also applies to over vehicles that most people forget they can also check up!
Checking the history of a van or plant is also a very good idea. In the vast majority of cases a seller is legit and buying a car goes through fine, but being caught out by car crime is not a risk worth taking.
HPI Checks, Car History checks and checking through log books (V5 Documents), service history documents and even looking up the garage that last MOT'd the car/van/plant is certainly worth doing.
Car mileage's can be verified, refurbished engines should be approved and also without checks you may purchase a vehicle that has been in an accedent and suffered damage that means it is unsafe to drive. This can be very well hidden and hard to spot even with a thourough driveway inspection.
Car history Checking services use the numberplate or the VIN (Vehicle Identification Number - which is/should be welded on to each piece of body work of the vehicle, including in the engine bay on show.) to check through thier database for information on the car - with these the more you pay the better the checks. Services offering to do it for a few pounds can work well but occasionally miss problem vehicles.