"What could be private in a relationship?"
letters addressed to one person and not another, texts sent to one person's phone, e-mails sent to an individual's address, feelings, thoughts, online activity, conversations with friends
I value my privacy very highly and I expect my DH to trust me. Because I respect and love him I don't do things I shouldn't.
I would never open a letter addressed to him, even a bill, I wouldn't read his e-mail, I would not check his phone, I wouldn't eavesdrop on a conversation he was having with a friend
He knows some of my passwords, but if I found out he had used them to spy on me on would be FURIOUS and feel utterly betrayed and I never use his passwords unless he asks me to. I haven't bothered to change my passwords on the basis that I trust him not to misuse them and he can trust me not to misuse his.
When people snoop because their partner is treating them in a shitty way I have a hard time condemning them for it, but basically I think if you need to be spying on someone you already know you have a big problem to fix.