this is a very dynamic topic and obviously differers from relationship to relationship.
i am lucky that i dont have the type of relationship where my gf feels the need to check mine or i have the need to check hers.
but if she wanted to check mine from time to time i wouldnt take it as a lack of trust, if anything i would take it as a sign of intelligence. anyone could act like an honest person and lie behinds someones back. yes i think trust is a core component of any relationship, but if my gf/wife checked my phone once every few months i wouldnt take it as a lack of trust. the things is to what extent and how often she would be doing it. if it was just once in a while to make sure everything was well it would be fine with me since i have absolutely nothing to hide. ive seen so many people be burned on trusting someone else that i think its kind of crazy that 2 minutes of a phone/internet background could save you months/years of a bad relationship. i said before i have the type of relationship where i ask her to check my phone/email for things and she does the same for me, so its pretty easy when both of us basically invite each other to check each others accounts (and this is because we trust each other and have nothing to hide). so for us its really no big deal.
i dont see the harm in it unless its frequent/obsessive. if neither party has nothing to hide then it shouldn't cause any harm. another thing about this is that some women/men dont understand when they are crossing the line. ive seen too many women stringing along men not realizing they were friend zoning someone and just thinking that the guy they were hanging out with were just friends rather than a guy who was desperate to get in her pants. once again i dont think this is a matter of trust since the girl could have the best of intentions here. this is a situation where a bf/husband/so should be able to step in and say "hey i dont think this guy has good intentions, stop hanging out with him", and that should be the end of it. this is another scenario that once again has less to do with trust, you could trust ur gf/wife/so all you want but that doesn't change some other guys intentions. so if ur gf/wife/so was serious about your relationship she should understand your concerns and stop. and the reverse could just as easily be possible with a guy stringing along a girl.
i guess my overall point is that people throw around the whole "trust" thing too much. i dont think "trust" should get in the way of common sense. should two people in a serious committed relationship trust each other? absolutely. but everyone is susceptible to mistakes, misunderstandings, and unclear communication, so trust isnt an excuse for ignorance. you should know if your wife is seeing a co-worker guy 5 days a week everyday at lunch, and ur wife/gf should know if a girl is texting you at 11pm with something like "how are you i just thought about that joke you said earlier today hehe lol, wanna see a pic of what i'm wearing?". if there is really nothing to hide, then why fault someone else for looking as long as its reasonable?
