Hi,
As my car is currently for sale privately I have taken on a gestor to deal with all the paperwork so it's all done correctly. What he has said is that the purchaser of the car needs to provide their NIE, passport and pardon or house deeds for him to proceed . He will draw up the sales contract which both parties will have a copy of.
As I've just bought a new car and needed insurance I went to my brokers who arranged for cover so I could collect the car and directly from there I went and paid. If you pay by card or bank transfer that saves you going into the office. Also dont accept the first quote as most companies will give you a higher price. I had a quote from my broker then another from Sabadell which was cheaper and 150€ less excess so went back to the broker who promptly gave me a cheaper quote and 50€ less excess, all with the same cover.
Hope this helps and you get what you're looking for