Any road taxes, fines, accidents etc incurred by the driver of your car will be in your name, and you will face the consequences! Imagine what could happen in the case of a fatal car crash, charges, compensation claims etc! It could go on for years.
Apparently it is common for garages not to do the transfer until someone buys the car, as it saves them paying the transfer fee; I don't know if you can insist that they do put it in the garage's name-possibly if you paid the transfer fee;might be worth it for peace of mind. It is not aa good idea to take the trader's word that he "will do all the paperwork" as he may never do it.
Apart from that, I think you can go to a solicitor or gestor to do "a baja" so that you are not responsible for anything that happens regarding the car.
You could contact Graham Shelton who writes in Costa Blanca News, who will advise you-just Google it. or I'm sure Johnzx will enlighten us!