Of course, I detrash my own cardboard! Plastic wrap goes in the right bin, as does hard plastic. Non-recyclables go in the trash bin. When I fill up a repack box of its cohorts, I put a topper on it.
I think most TMs at my store are pretty good about this, but we do have one or two who just don't seem to get it. And a few who seem to think they're special and don't have to break down their boxes before tossing them in the baler AND then don't crush it.
Even our vendors are cooperative on this, at least the ones I've seen taking care of their trash the same time I am.