The turks access your data (the questions) through Amazon, and I don't see how this will happen without Amazon having access to that data.
https://docs.aws.amazon.com/AWSMechTurk/latest/AWSMturkAPI/A...
This seems one of the more popular ways to do turks. You send people off to your own site, to show them stuff. All you really provide is a url, like www.mysite.com/questionId=123456... and render whatever you want when that appears
If you're really concerned about a third party having access to your data, you could just host the survey yourself.
(A related question: would it be possible to use Amazon Turk to train an algorithm that answers Amazon Turk questions?)
Then you would only accept the submissions that have passed the attention checks, and pay those mTurkers.