That wasn't the question. What do you do when you did read something inadvertently? You can't unread "Irregularities in the pension fund". Do you pretend that you don't know? What if it's something illegal / against company policy / unethical?
We used to call it 'being trustworthy'. Not sure what the term is today.
People need to know that they can rely on you under pretty much any circumstances, otherwise they'll stop calling and you won't be able to do your job. That means ignoring pretty much everything.
I say pretty much, because there is a line past which you cannot remain silent. For me, it was child pornography on a customer's computer. I called the police and handed over the equipment.
This was in a small town, and it ruined my life, by the way. The owner of the computer was a prominent citizen who immediately accused me of planting the material, then began a slur campaign against me. The town, as the saying goes, wasn't big enough for the both of us. After more than a year of this, I had to leave. I'd lost my job, and I'd lost half my friends.
Some time later, I ran into an acquaintance from that town in an airport. His first bit of news that that the kiddie diddler had finally been convicted. His own smear campaign finally had the effect of bringing three adult victims of his out. They testified against him and put him away. The lesson I learned is that, sometimes, there is justice in this world. But it doesn't come free.
So yes, you need to be - and you need to be seen to be - completely, implicitly trustworthy. How you do it is simple enough: Always be there, never be seen to be part of the gossip. Be open and obvious about everything you do, and never, ever work in someone's office with the door closed. Equally, though, you need to be seen to be the kind of person who will do the right thing. That's a little harder to do and, as I've recounted, sometimes comes at a cost.