It is handled by Linden Lab, hence the transfer fee that will be paid by either the seller or the buyer.
The way it works typically for me, at least the last dozen times, is I contact the seller and get the name of the region and their name, and put in a help ticket under the region transfer category indicating my intention to buy.
I then give the seller my ticket number, which they add to their ticket in the same category with intention to sell (and typically the intent to cover the transfer fee), and they send me their ticket number.
I then add that number as a comment on my ticket, just to help LL out.
After that, LL handles all of the monetary exchanges and the billing alterations.
The region will keep its tier date across the transfer, so you will be on the hook for tier when it is due. Which is why most region sale ads contain the tier date.
All in all, a very secure system. And in some 20 region purchases from other residents I have never had an issue.