Hmn, "background check", that's a bit vague and doesn't describe what kind of information is actually gathered. There is an industry of 'private investigators' who actually don't do anything other than gather information from public sources. Given that the words "background check" don't accurately explain which information is actually gathered, I don't see how we can answer your question on whether the contractors needs to know.
But, my gut feeling says you're going in a wrong direction here. The classical, time-proven, non-offensive way of learning more about a contractor is to ask for a list of references -- and actually call them & ask good questions. That's what I would generally suggest to do instead.