The nursing school I'm attending, BHSN, did drug tests prior to the start of school. After receiving our acceptance letter, admission was dependent on passing the drug test, along with all the other medical stuff (vaccines, etc). They gave us about 2 months notice of a 60 day window wherein we could have our drug screen done. I don't know of any clinical sites that do screenings during clinical, though it's always possible!!
That said, my nursing school is a hospital-based program and all of our clinical sites are all linked to the school.