I wouldn't say 'check' as in snooping/peeking as we are open books to each other and have no secrets. (unless it is a gift).
My email opens right away when this thing is open so he can read whatever he likes and I check his email in case there is something important since he can go weeks without being on the computer. I take care of a lot of his Facebook stuff since he doesn't care too much and would rather read a book. We share the same passwords- not much isn't known.
People who do this in fear that their SO is receiving inappropriate texts have a level of insecurity that broaches on the immature IMO. How would checking texts prevent one from acting on it if they really wanted to anyway?
There is a saying 'you create what you fear' and if you act suspicious all the time checking and questioning every move your partner makes you will drive them away if you are constantly behaving like this.
If a person steps out of the relationship for affairs that ceases to be my problem and one the other gets to live with. I am not a jealous person anyway and if my DH walked towards another then I suppose it would be time for him to leave if we couldn't work it out.
But it would be his working on it more than I if the marriage was healthy (IMO a healthy marriage doesn't have mistrust). This is not on my radar in the least LOL.