If you allow someone to test drive your car without checking they have a license AND insurance you are committing an offense. As Mike says above, you take a calculated risk even if the above is observed. If they prang the car then you will almost certainly be on third party insurance and lose the value of the vehicle you are selling. I have sold a few cars privately and most of the transactions have run smoothly. Be very aware if more than two people turn up to view the car.
Trade-in values at garages are on the low side so you need to weigh up the advantages (more money) with the possible disadvantages (Test pilots, who have no intention of buying the car - time wasters). Hassle free option is to trade-in. Some potential buyers are very nice to deal with but you must expect the pain in the ar$e buyer who just wants a bargain by trying to pick as many faults as possible. If you are an honest seller, then list all the faults you know of and show the potential purchaser. Be very wary of cash. Someone I knew accepted cash and after the purchaser left the hundred pound bundles were short. Also be aware of counterfeit money being handed over. Bank transfer, as Mike suggests, is usually instant and can be verified if you have internet banking. It all sounds very scary but you need to be made aware of what can happen.
Has anyone on the forum actually sold their car to Webuyanycar.com or similar? Your experience would be helpful if you are willing to share, as Herbie did with his Lexus purchase.
Just my thoughts, others may add more.