On that subject, I (a European) went to the US last week. It was time to pay at a restaurant in their needlessly complicated way where they hand you the bill, you return them the bill with your card and then they return once more for you to fill out the tip. Shortly after the second step the waiter returned apologizing, saying they could not bypass the PIN on the card like they normally could (which was slightly startling to me) and asked if I could come with them to enter it on the payment terminal myself.
Must have been a debit card that flat out refused anything but Chip & PIN.
Most cards aren't like this.
With my UK issued AmEx & Visa cards (both Charge/Credit), at certain places terminal didn't even ask me for a PIN, and the transaction just went through as "Chip & Signature"
A rogue terminal can decide to authorize the transaction with a “signature” (there are legitimate uses for this)
Or even with no PIN at all (there are also legitimate uses for this)
It’s also possible to do either of these 2 things and then report back that the transaction what authorized with a PIN