I've got the Amazon Appstore app on my UK Nexus 7, but I was unimpressed. Firstly, it doesn't detect which apps are already installed, which is basic functionality surely? So stuff like Temple Run - a free game I installed via Google Play - just appears as a game you can "purchase" for 0.00 quid (since when is a zero-cost item a "purchase"?).
What's also bad is that if you download a free app, it'll not only try to overwrite the already existing (probably identical) one, but also e-mail you a receipt for 0.00 quid by default - how double dumb is that?
I also don't like that any free downloads (including app of the day) insist that you tie a credit/debit card to your Amazon Appstore setup - a sneaky trick to make sure that you go for impulse paid purchases later on and have your card info there ready to be debited.
Still, I've installed a free app of the day notifier free app (confused? :-) ) that whilst US-only at the moment (I've e-mailed the dev cos it needs to support other countries including the UK now), will notify you each day about the US free app, but show the UK one when you click on the notification, which is good enough for now. Won't be using the Amazon appstore for anything other than the free app though, especially since I've still got most of my 15 quid Nexus 7 credit left on Google Play that the Amazon Appstore doesn't accept of course.