It's interesting how broad and disjoint the US Federal government is. I'm a Federal contractor and while many of my coworkers are not US citizens, the government does do a quite thorough (non-clearance) investigation on all of us that involves filling out a rather long form, providing past addresses and contacts for each of those addresses, etc. AFAIK they don't do in person interviews for non-clearance background investigations, but they do send out questionnaires by mail to all of the people you list on the form.
But it was the federal government that insisted that we be US citizens. As far as I know, they didn’t require us to pass any special background check. And they didn’t require us to have any special training. But we absolutely had to be citizens (not just “legally able to work in the country”).