yes should be transparent of every thing. because you are husband and wife should know all about your doing.  It doesn't mean Just because you have the access of his/her email you have to get involve but you are looking his stuff.  It should be reciprocal. You should be sharing something if your partner also did share it. People are happiest when they have a match. You and your partner should be a match in that respect too.
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yes.
and let us not forget the congruent responsibility of that privilege or right