It’s hard to say what exact protocol those apps use. Amazon’s AWS is a RESTful service so it’s likely there app probably does that. As for PayPal, I’m not familiar with their API.
But REST is a great way for Corona Apps to talk to web services. network.request along with using the json.encode and json.decode functions allow you to easily move Corona Tables into data that can be consumed with web services since JSON is typically supported. Likewise, it’s pretty easy if you know PHP to make your own RESTful API to get data from a MySQL database and get it into your app. Simply json_encode a PHP array on the server side, return that json string to your app using network.request and then use json.encode and then you have a Lua table all ready for you to use.
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