Behind closed doors, powerbrokers across the world in politics, tech and media have convened at an annual in-the-know retreat hosted by Peter Thiel to rub elbows and examine hot-button issues. Dialog, the invite-only society that bills itself as a forum for leaders to “discuss topics off-the-record,” maintains a low public profile and has never shared a list of its members. For decades, little to nothing has been uncovered about the group.

That changed earlier this week, when a Swiss independent researcher leaked information about the organization that was left open in the code of the society’s website.…