I don't generally open DHs mail. He doesn't open mine but occasionally we do. We're not offended by it. In terms of passwords....I don't use his computer and he doesn't use mine. I have a fair idea of what his usual passwords are and he does know mine.
But we trust each other. So there's really no need to look through his stuff or vice versa.
If I'm home and mail comes that looks important I'll tell him I'm opening it. But usually I just put it on his desk.