The rumour at Mesa is that they will get some steam-gauge 737-400s and fly them for DHL. Not anything about flying for Amazon.
Mesa flying for Amazon is a rumour of somebody-else's creation, presumably because people want to hate on Amazon but not on DHL.
For Mesa to fly 737-800s for Amazon, Amazon would have to take aircraft out of its existing contract with Southern. Southen currently has a contract to fly 15 of them by 2021 -- basically 5 per year 2019, 2020, 2021. The initial 5 are already in the hands of Southern; 3 are flying, 2 are not yet in service (last I looked both at VQQ).