Do you find yourself deliberately ignoring office politics, overheard conversations, open documents or emails, etc as you go about your work?
If not, how do you preserve the impartiality/neutrality which seems (to my novice mind) necessary to be effective in this position?
In either case: how do you deal with the possibility of accidentally learning something you're not supposed to know? E.g. troubleshooting a user's email program when they've left sensitive/eyes-only emails open on their workstation. Are there protections or policies that are standard, or is this a legal and professional gray-area?"
Let's start in order: Yes, I ignore office politics. I'm there to do my job.
No, I don't ignore overheard conversations. I listen in to learn what's going on behind the scenes and make mental notes for future use. Offices with women are generally the best for the amount of gossip you hear, especially about other women.
No I don't ignore open documents or emails. Does that mean I root through things? Of course not, but if you leave it open and I'm in your office waiting for some poorly coded software to install/upgrade, I'll be nosy and look at what's open and, like above, make mental notes. It's fun to read some of the HR documents, especially when people are getting written up.
As to impartiality, I'm there to do my job. Whatever I hear or read I keep to myself until such time it might come in handy (I'm so waiting to win the lottery so I can let loose on things I've overheard/read).
To your last point, you're there to do a job. Do it. What you read or hear stays with you. You don't say a word to anyone, inside or outside the organization about what you came across.
Generally, organizations have policies in place which state you can't reveal inside information to the outside world so if you come across something sensitive, such as personnel matters or impending mergers, you can't say anything. There is no gray area.
If you're familiar with the show Hogan's Heroes, you are like Sergeant Schultz: "I know nothing!"