HR usually conducts them, using a carefully orchestrated, one-sided protocol. Yes, there are legal issues, and these must be addressed. But it's the hiring manager who should conduct these checks, after being taught how to do it right. A reference call from one manager to another is very different from a call from an HR rep. Managers can delve into more detail, and they have both the expertise and the prerogative to pursue lines of questioning that HR lacks.
Peers are more likely to be open and blunt with one another.
While the answer matters, it's the hesitation or the enthusiasm of the respondent that's critical. Manager to manager, this one question can reveal more than any other kind of reference check. When you're hiring, don't pay lip service to the importance of reference checking. Involve the people who will work with the new hire. Headhunters know something that job candidates hate, and that most employers are too busy to think about.