Depends a little on what you run on the Kindle. It's perfectly possible to run 100% open source. Amazon isn't exactly making it super convenient, but they're doing very little to prevent it. Stock, of course, they retain the ability to do so but on the other hand A) sponsor the hardware and B) provide updates and very good support.
I consider it a pretty fair tradeoff. They're not stopping you from churning your own butter, so to speak, but do count on you deciding it's easier to just run their stock stuff. Like a lot of semi-locked hardware, it's not like they'd be such floozies about proliferating the hardware if they weren't making a lot of it back in book sales.
(Edit: It's never hooked up to the Internet, except when plugged into my computer, so I don't see how Amazon or anyone else is going to be able to touch it.)
If the answer to any of these questions is "yes", then no, you do NOT have complete control over the device.