From what I can remember the seller only needs the log book and passport. A residence permit wouldn't hurt. The buyer needs passport, residence permit, driving licence, tax number and an e-paravola from the bank proving that the fee has been paid. The bank will need to see the log book to decide what fee to charge, since it is based on engine size.
I can't guarantee the above. The last time I gave away a car and bought another KEP did everything for me. I just went to the bank to pay the transfer fee for the one I was buying. When I bought the motorbike I went to the vehicle centre with the original Greek owner and he did all the talking. I just handed over whatever they asked for. In theory if you are going to do it at the vehicle centre you don't need the e-paravola. They have a machine there that allows you to make the payments but it is often not working so I would recommend getting the e-paravola from a bank anyway. There isn't a bank close to the vehicle centre if the machine isn't working.
KEP at Vamvakapoulo are helpful as is the one on the road near the vehicle centre, i.e. on the right hand side of the V formed by the two roads that join the Souda road at traffic lights not far from the IKA building with a SYN.KA in the middle. I agree that the one at Platanias, actually under the townhall at Gerani, is useless. Whatever you ask they can't do it.
As far as I am aware you can pay the e-paravola at any bank. Many years ago, before the e-paravola existed but there was an equivalent multi-part form system, you could only make this type of payment at the National Bank of Greece but that has not been the case for a long time.