Not unless I need to. LOL
I mean... generally, I don't care about his wallet. But sometimes, when I am balancing the checkbook, he'll be gone running or something, and I'll check for any gas receipts he might've forgotten to give me.
As for mail, not much comes address specifically to him alone. Most of it is to both of us, or to "homeowner", or it is an obvious bill that happens to be in one or the other of our names. Most of it is in my name, but some of it is in his. He doesn't pay the bills, I do. So I open those. And good thing I do, too, because he knows they are bills and so he doesn't open them.... just gives them to me.
He won't go in my purse. If he is walking out the door and asks if I have any cash, and I say, "yeah.. there's cash in my wallet," he'll stand there right in front of my purse and say, "well, will you come get it for me?" I don't know if he was just brought up to respect a woman's privacy with a purse, or whether he just doesn't want to dig through it to find my wallet and search for cash. ;)
We don't really have dressers in the bedroom. We have built-in drawers in the bathroom. And mostly all that is in his is clothes (underwear, Tshirts, socks, shaving stuff/contact lenses). Since I put those things away, of course I open them. He opens my one drawer that I keep spare razors and toothbrushes in. Otherwise, he doesn't bother with my drawers.
Would I care if he did? Only if he came across the birthday card I bought a month ago and stashed until time to give it.
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ETA: If you are really interested in "snooping" then I am surprised you didn't mention cell phones. That seems to be the primary "snoop" issue for people. As for phones, I use his, he uses mine, we don't care. (well except that he puts mine down where ever he was when he finished instead of putting it back so I know where it is). We have nothing to hide.