Yes, but it's not about you. It's about your company, the legal obligations they have to protect your data, their customers data, and their internal IP from outside intervention.
Consider that you have a legal obligation to protect the privacy of your data. Now if you are served a warrant to give up the data or to secretly tap connections to your server, you are probably legally covered. But when they just go straight to Amazon to get your data you have no such cover. Users can rightly ask why you put their data in a place out of your own control, your own legal accountability.
Prior to this year, one would have said that such events were so extremely rare, and in such exceptional circumstances (on credible threat of an immediate major terrorist attack, etc) that it was an acceptable risk for a business to take. That's now flipped. The presumption is that if you store data in the cloud then it is being routinely intercepted and is freely available to a multitude of parties outside of your control. That's the crucial change that is going to put cloud storage and services off the table for anyone with sensitive data.
Until the industrial espionage and insider trading leaks come out. Then all hell will break loose.
Or they will still not care.
If that's something that affects 0.001% of business, and if some culprits have been found and punished (for the leaks to come out), then businesses will probably continue using AWS and not caring, thinking "what are the chances"?
Except if something causes direct and measurable money loss, they wont really care, even if it's proven to have hurt this or that other company.