They don´t check your ESTA physically but many do ask at check-in. And I had a flight delay due to the fact that another person did have ESTA but had the same name as someone on the no fly list of the US authorities. It did take quite a while to sort this out and then the person was able to fly (and we too).
Exactly, the airlines have to transfer the passenger data to the US and then get the o.k. by the US authorities prior to departure.
steamboats