I have a password in my email and I've never been asked for it neither I asked for her password. My mobile is used by both of us cause I have a monthly payment and she has a pay as you go.

If there is a trust the problem or even just the question would never occur. If there is a bit of a doubt it's not worth it to live together, I don't like wasting my time on private investigation while I can do something more interesting and useful...