Mine is easily accessed through my computer but I’d be pissed off if I found her reading mine. And I wouldn’t read hers.
I leave myself logged into yahoo mail and he leaves himself logged into gmail on our joint computer, and we know each other’s passwords anyway, from when one or the other of us is too lazy to get off the couch to check our own email.
I would feel uncomfortable checking his email without his request, and I don’t specifically read his email, but if I notice a few things when I open the webpage, I don’t think it’s a big deal. I feel the same way about my email - if he was specifically reading it, I’d find it a strange thing for him to do, although I wouldn’t feel offended or hurt. There’s nothing in there I don’t want him to see.