There's a lot that the Amazon Fire Phone does different, which is no surprise given that while it's an Android-based device, it's mostly gone off the reservation with its on suite of apps and services. But one area in which Amazon hasn't changed a thing is screenshots.
To take a screen capture on the Amazon Fire Phone, all you need to do is simultaneously press the power and volume-down buttons. You'll get an audible alert letting you know the screen shot was taken — same sound as in stock Android, actually, and the Fire Phone screen capture will be saved to the phone's gallery, in a "Screenshots" folder.
And just like in stock Android you don't even need to go into the gallery to do something with that screen shot. Pull down from the top of the display and you get the usual "Screenshot captured" notification. Pull down on it and you'll have the option to share the screen shot with any connected services.