Who is able to change details after selling a vehicle? I assume it has to go to Trafico in Alicante.
Any good suggestions locally for price and reliability?
Yes, you have to go to Tráfico in Alicante, having previously made an appointment on their website. Before that, you need to make out a "Contrato de Compraventa" (you can download a blank one from the internet) which both parties sign. Then you need to establish the value of the vehicle for tax purposes. I used a website: http://gestiona.madrid.org/vlin_valora_bienes/html/web/VTurismos.icm?ESTADO_MENU=1_2_1
Alternatively, you can go into the Hacienda office in Alicante (quite near Tráfico) and they have got a computer you can use to establish the tax value of the vehicle. Then you get the relevant 'Modelo' and pay the tax at your bank. Then you go back to the Hacienda with the receipted copies of the Modelo and they give you a certified document to take to Tráfico. You turn up at Tráfico at the appointed time and pay the fee (52 Euros, I seem to recall). You are given a 'turno', but there might still be people in front of you (41 people the last time I went!). When you eventually get to the counter, they give you a new 'Permiso de Circulación in the new name, providing the paperwork is all present and correct. I just took everything I could think of that they might ask for, original and a copy of each.
It's really satisfying when you come out there with the new permiso. You feel like you've just passed your A-levels!