I am not sure if this is law, but considering background checks aren't cheap, i'd ask for the candidates consent. The first reason is to be covered by law, the second reason, if the candidate refuses to grant consent or vanishes, I just saved myself a ton of money, once on the background check itself, second on having a potentially bad employee.
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.