I never look at my husbands phone, and he never looks at mine. I trust him enough to believe that he's not texting a secret lover and I wouldn't particularly want him looking at the texts between me and my friends.
If his phone goes off and he's not in the room I still wouldn't look at it. I would just tell him when he came back in, and if it rang I wouldn't answer it either.
I really don't think there is a good reason for reading your partners texts/answering calls.