This is what I've done in my HD 8.
Amazon apps update themselves. Can't really do anything about that. If the app update requires a permission change, it will tell you that it needs a manual update - so you can confirm the permission change.
I have Google Play Store on it too, and turned off auto updates through it.
Google Play will show some apps loaded from Amazon as installed, but won't bother asking to update them if one comes to the Play Store before they hit the Amazon App Store. So the two stores generally stay out of each other's way. It's weird that the Play Store will only show some apps from the Amazon Store, like Hulu or IMDb, but not others like Netflix or Wikipedia.
However, the one app that gets pegged by both stores with updates is the Alexa app... strangely enough. The Play Store immediately wanted to update Alexa upon initial install. I ignored it for a while, but one day hit "update all" by mistake when there were a bunch of app updates at once in the Play Store. Alexa still worked. A few weeks later, Amazon wanted to update Alexa, and listed it as "side loaded". I let it, but later said it failed. But Alexa still worked, and the App Store no longer said there was an update. A few days later, Play Store says there's an Alexa update. Went ahead and let it update again, and all still works.
Maybe one day it will all go haywire, but it's okay for now.