Ohio = no, although some counties may do an emissions check.
I lived in Missouri, where vehicle inspections were required. But you can always find someone who will pass it for a few extra bucks. Although they do have undercover inspectors that go around and check to see if an inspection place is doing inspections properly and legally. I briefly managed a repair shop, and when we got our inspection license, we were talking with the state inspector, who told us the previous owners (before they shut down) had been under investigation for selling inspections stickers illegally. Or passing cars that shouldn't.
I even had the owner a used car lot down the way approach me about inspecting their cars. Could send us business, if only we would fail a few minor items wrong on the car, and pass everything else. He was willing to offer a kick back on every car sold then inspected and passed. I declined on his offer, as we ran a clean and honest shop.
So I can bet you there are some unsafe vehicles running around in states that require inspections. maybe not as many as in states that don't inspect, but they are there.