Of course it's in secret and simultaneous.
Also... the "reliability" of safe thought problems doesn't translate well to real life. In thought problems, almost everyone says they'll flip the switch and let the trolley run down one person instead of five... but in functionally similar situations in reality, people freeze and don't flip the switch at all. There's a massive difference between making a noble assumption based on pure logic when you're safely sitting on your couch enjoying a beverage in front of the TV and actually being required to take an action in reality when your life is on the line.
There's no way to prove it, of course. But I believe that if there were a functionally similar real life situation, where there was a choice between completely avoiding an obvious and advertised risk, and placing yourself at risk of death... less than half of the people who say they'd pick blue would actually pick blue.