The Amazon App Store API (https://docs.coronalabs.com/plugin/gamecircle/index.html#general-functions) does not have a method for logging a player in, it is missing an API call. The popup that appears does not consistently display either.
I’ve created a Feature Request to describe the requested functionality: http://feedback.coronalabs.com/forums/188732-corona-sdk-feature-requests-feedback/suggestions/15161550-add-support-to-amazon-game-circle-for-signing-a-us
I’m not sure a sample project is worthwhile since there is no bug to demonstrate, but if I can provide anything let me know.
Thanks,
Albert