It needs either the mini hub or your phone to be home, but your phone doesn’t have to be home.
The mini hub (which is really small, about the size of a deck of cards) has Bluetooth and Wi-Fi. It communicates with echo cloud to cloud, same as smartthings does.
What’s new is that these are the first third-party buttons (not sensors) which can be used as the trigger for an Alexa routine. It is a cloud cloud integration, but it’s still very cool. 
The mini hub does, like all certified HomeKit bridges, work locally with homekit. So there is that potential, I just don’t know exactly how everything works with the new SDK.