They are certainly not as clean as a modern diesel but if they are properly adjusted they aren't bad. Most of the smokey ones probabley haven't been maintained for years.
This includes not only a visual inspection of all safety-relevant parts of a car but also a check of the emission values.
How is this handled in the US?
There are exceptions with older (vintage) cars, and relatively new cars.
As with most things, these laws vary from state-to-state, and even county-to-county within said states.
Kentucky had a limited emissions inspection regime from 1999-2005, when it was killed by the Republican governor. The only current inspection requirement I know of is if you purchase an out-of-state used vehicle. We have plenty of ancient cars on the road that have never been inspected.
Sometimes police officers will pull you over if you have obviously-broken headlights or such, and tell you to fix it, but not often.
en.wikipedia.org/wiki/Vehicle_inspection_in_the_United_States
Source: I have an appointment for car inspection this morning.
Edit: certain counties