Tough question to answer. As variable as the terms "BF", "FB", "FWB", "LTR", "Partner" and so forth.
I've always reached that point when I'm living with the guy. If we've gone that far, then it assumes a level of trust, and in my case of love, such that personal boundaries should be dropped.
Both my late partner and my present shared our online world, all of it. I even get his calls mirrored on my own phone, and Apple Watch, to ensure he hears them. And sometimes to answer those I know he wants picked up immediately.
I actually turned off his email mirroring - it was too much for me! Along with his texts, which are both high volume. But we have no secrets. He can see what he wants, although for myself I tend not to want to pry.