My first thought is to start converting any and every thing that can be to a electronic bill or notification vs. paper. If your DH won't do it, do it for him. Use his email address and give him a list of the passwords if you have to set up a online account to do it. Even though it'll take time to do it, ultimately you gain from it as you won't have that mail coming in. If he is so paranoid about you seeing things then suggest that is the ultimate way to make them private as only he would see it in his email.
On the back of the credit card offers there is a toll free number to call to get yourself removed from all the major creditors offerings. It's not company specific--one call can remove you from a joint list. It's automated, so you can call and remove you and your DH in a matter of minutes. IIRC it removes you for a few years.
If he gets any bills necessary to the running of the house (gas, electric, etc.) or pertaining to joint financial obligations I would open those for my own sanity whether he likes it or not.
I agree with PPs that you should have one designated storage container for his mail. If it moves to other places in other people's way then it's shredder fodder IMO regardless if it's been opened or not.
I'm a confessed pile maker. I also open ALL the mail for our house because even if DHs name is on it, I still pay the bills. I have found the best way for me to limit piles is to cut down the incoming mail as much as possible. I stopped subscriptions to any magazines I didn't read within a couple days of receiving. I automatically toss catalogs in the recycle bin because all that stuff is online anyway. I converted all the bills I could to ebills either through my bank's bill pay system or through the company themselves.
Good luck!