If I was seriously bothered about something then I probably would check his phone. You then have the dilema whether you say anything about what you saw or if you choose to leave it.
I trust Mark so I'm not bothered about his phone, e-mails etc. He has told me his password before but it's a complicated one and it just goes straight out of my head. If we are sitting in together, he'll happily check his e-mails in front of me so I know I have nothing to worry about. I don't think twice about asking Mark who it is if his phone beeps mainly because I'm so nosy!
I leave my phone around all the time, he knows my e-mail password and we are signed up to a few things where my address is the contact e-mail so I really wouldn't care if he went into my e-mails.
He goes in my bag to get stuff all the time aswell, no problem to me because I have nothing to hide. I also go in his wallet to get money etc and there's no problem with that.
If he were to start carrying his phone around and never leave it out of his sight I'd get slightly worried but he's always leaving it, even if he goes out.
I have a friend who checked her boyfriends phone ALL the time, she became a nervous wreck. The problem is that you only get one side of the conversation and it can be misleading. He did actually cheat on her but I don't know if he would have, had she not shown how much she didn't trust him and been so insecure all the time. Its a viscious circle. Once you start to lose your trust in somebody it's very hard to get it back.
As it happens, my friend split up with guy and he is now with my sister and they are head over heels in love with each other! I don't agree that 'once a cheater always a cheater'. You can get over it and it has a lot to do with how happy you are in a relationship.