Whether it's a girl or guy, that is not a good start to a relationship. A person shouldn't act a certain way because someone else wants to control them. Who knows what other privileges that person will take from them. I could see where some person may think that is the only way they can be considered "cool" or keep that person happy, but this is only harming the person who says that.
Sounds like very immature people at work - that would think such things would be good, in the first place. Even very intimate relationships include amounts of personal space for each individual, and to try and negate that is just plain silly and vastly insecure.
There is such a thing as being too "clingy" in a relationship, and it's a bad thing, plain and simple. To try and enforce rules on both people, shoving them in that direction, is foolish.
I hear so many people who are supposedly in happy relationships come out with things like "He/She has all my passwords to everything, he/she can look down my phone whenever he/she wants". How can that be classed as a happy relationship when there's clearly no trust there. The fact that either of them feel the need to check their partner's phone, Facebook or whatever it might be (whether or not they're given permission to do so) demonstrates they are insecure about the relationship. I've been with my boyfriend for nearly 6 months and never once have I asked to go through his messages, because 1) I have no right to and 2) I trust him. Simple as. And I would be offended if he ever asked to look through my phone (which I know he wouldn't). If I ever feel any worries or doubts about the relationship I speak to him, and vice versa. No relationship is perfect, but going through someone's personal messages (with or without their consent) is not the way to get through your problems.