Only available in Spanish unfortunately, but the instructions can be found here:
http://www.dgt.es/tramites/vehiculos/pdf/Transferencia2.pdf
and the form to be completed here:
http://www.dgt.es/tramites/listado_impresos/impresos/9.050-Castellano.pdf
It's a long time since I sold a car privately, so can't remember exactly what I did, but since it's important to make sure the car is registered to the new owner so that any subsequent tickets / fines etc don't land on your doormat, I think it best to notify Trafico (in person) of the change of ownership yourself rather than leave it up to the buyer. Of course, that means you will pay the transfer fee, but from what I can make out it's only €43.40 anyway so worth it for the peace of mind.
If you are selling to a dealer, they will no doubt tell you they will handle all the paperwork. It's easy to think that once you've got your cash you can forget about it, but I would still want to be sure that it's no longer in my name.
If anyone has recent personal experience, perhaps they could add to this?