IME, HR does background checks and uses a 3rd party to validate employment history and education. Usually hiring manager or recruiter does reference checks.
At my workplace, we do employment and education verification. We don’t do criminal background checks because it isn’t strictly necessary in our space.
The last time I had a background check, despite providing the name, address, and phone numbers of past employers to the background check firm, the resulting report indicated in several cases they had instead called unrelated companies in unrelated industries in states I had never lived or worked in. There were no repercussions when those companies (understandably) had never heard for me. Presumably, HR had checked the box that the check was performed, but they didn't actually care about the result.
One of the cases they flagged as "unable to verify employment" was a stint at a company I had founded, where I'm still listed as the LLC manager on my state's business registration website, i.e. a simple Google search for company + my name would have sufficed. (The name of the construction firm in a neighboring state that they repeatedly called instead wasn't even similar.)