I'm hoping to set up a mailing address that is completely separate and private from prying eyes at home. Will USPS send anything (e.g., billing reminders, address verification) to the home address I am required to keep on file with them? Or is there any other reason anybody would know that I got a PO Box unless I told them?
No you're good. You can sign up for the po box reminders by registering on their site.
So you're not freaked out, when you go to pay for the box, they will ask for ID and some other documents to prove your identity and your address. Some are like DMV clerks and other sign an X for their sig. But the scrutiny depends on where you live.
Edit: come to think of it, I think you can be pretty granular in your PO box settings. Ask the clerk, they are usually helpful and friendly and hand me my packs with a smile everytime.