If we are asking about Overdrive, they always send me to amazon.com.
Then, on the same page where I say which family Kindle should be sent the book, I am asked to purchase a group of "Featured Recommendations."
When going to return, Amazon suggests I buy the library book. The only way I'd ever do that is if was a paper one I'd lost
Ideally, Amazon should be paying the library (or at least Overdrive) for the opportunity to pitch its products.
Is there any definitive answer at to whether Amazon either gets paid, or pays out, when I borrow a Kindle-format eBook through a public library Overdrive collection?