In Texas, if you pull a background check on someone you're required to provide a copy to them which I always felt was a good minimal standard - at least then you know that someone pulled the check and what they found.
EDIT 1: A quick look shows a guidebook for employers which references this, but the link to the Texas state code talks about in-home employees. https://www.twc.texas.gov/news/efte/references_background_ch... I'm still unsure about the law related to other kinds of employees.