You can buy the details from DVLA for �5 if you can show good reason for needing to know - anybody can.
Good reason includes the vehicle has been abandoned on your land or you need the driver details to pursue a claim in civil court that your insurance company doesn't cover.
I have done it a few times but you will be committing a criminal offence if you lie to obtain the information. I have been able to provide a reference number from the council, the police or the insurance company to back my enquiries although this isn't strictly necessary:
http://www.direct.gov.uk/en/Motoring/OwningAVe hicle/AdviceOnKeepingYourVehicle/DG_4022066