From AlexUSA:
"Next, you will go to the vehicle registration office and transfer ownership in the "green book" (ownership document that comes with a bike in Thailand). You don't need to bring the old owner with you."
You can do this ONLY if the previous owner has provided a "power of attorney" form. These are available from the registration office, and will enable you to transfer the registration by yourself. If not, he will have to be present.
There will be a charge for the tranfer stamp at the registration office. This charge depends on the age and engine size of the bike. Should not be too much on a scooter.
As far as being on a tourist visa, my first bike here was bought that way over 10 years ago. I used someone's house for my address and got the form at immigration. Back then it was 100 baht in "tea money", now it is 500 baht at the immigration office. You will need some photos of yourself in addition to the proof of residency.