Too Lawyerly; Didn't Read: Mods are explicitly not agents of SE, and no one but an agent is supposed to access the private info mods do access.
The moderator agreement on each site says (emphasis added):
I acknowledge and agree that I am an independent volunteer moderator to Stack Overflow and I am not an employee, agent or representative of Stack Exchange Inc.
Well and good, that's standard phrasing.
We almost never share any of your personal information with non-agent third parties […] Other than on Careers, we won’t share your personal information with non-agent third parties unless we are required to do so by law, or if we believe in good faith that disclosure is reasonably necessary to protect our property, rights or those of third parties or the public at large.
And earlier, where it might clarify this, it specifically distinguishes moderators from agents again (emphasis added):
When we collect your personal information, we’ll tell you how we’re using it, any types of third parties to which we might disclose it, (other than moderators or “agents,” such as vendors or contractors, who are only processing such information for us or at our direction), and the choices we offer you to limit the use of your information.
This distinction is upheld by a recent answer by Tim Post, the Director of Stack Overflow Communities that opens by saying (emphasis added):
Moderators are not agents or employees of Stack Exchange.
Moderators have the ability to access any user's PII without needing permission from anyone else, and although this access is logged and isn't exactly routine, it's not anywhere near as uncommon as the second quote would suggest. Taking these all together, the policy appears to say that ♦ moderators are not allowed to do what, in fact, they actually do. That's quite distressing.