If you don't want Google, Amazon, Facebook, Twitter, and everyone else to know everything about you... you have the wrong phone. That ship has already sailed.
That said, I would assume the Amazon store is just doing a license check. That's what prevents piracy. You could get a bootleg .apk file from your favorite warez site and install it - but if it calls home and hears back "Nope, you're an unlicensed copy", it will terminate.
I'm pretty sure the same thing is happening with Google Play Store apps. If you rooted your phone and got rid of Google Play, I wonder if everything you purchased from there would stop working too?
BTW, the FAOTD today is "Draw a Stickman: EPIC", which has few permissions and many good reviews. I'm gonna snag it.