It depended on the kind of meal, the age of the kids, and sometimes the circumstances.
There were snacks at the table, individual meals someone may have at the table (like one kid eating breakfast when the other had gone to school earlier), and sit-down/family dinners and holiday dinners.
I taught them when they were toddlers that trash from meals/snacks always goes in the trash. When they were little enough I'd get the plate. When they were still fairly young they just automatically scraped and rinsed the dish and put it in the sink or dishwasher. (Learned by example I, guess.) I let everyone know that I wanted to wash the dishes, but washing a couple/few plates wasn't a big deal (and I didn't trust the cleanliness and soap-residue-less-ness" trusted to children). One thing they would do when they were young-but-not-young-preschoolers was to leave the small, disposable, cups we used for between-meal drinks on the kitchen table. They'd get rid of the rest of their snack or eaten-alone meal and leave a Dixie cup with, for example, juice in it on the table. They did that with the idea that they're come back for the juice in a little while, or they'd sometimes leave empties with the idea that they'd come back to use it again. (They'd already heard, "Let's not go through them like water," so they'd learned that much.)
With sit-down nightly dinner/holiday dinner, most often I didn't want them doing more than, maybe, getting rid of their paper napkin. More formal/formal-ish dinners, to me, require one person to kind of be in charge of the table clean-up and leftover "management". I suppose, at the back of my mind, I thought kids also need to learn how to just sit in and have a formal dinner without "everybody getting up and being a busy crowd" in the kitchen. So with sit-down dinners, some kids (I only have three) would bring their plate out. Others may. It all "just depended".
After all awhile everyone got old enough to feel uncomfortable about leaving dirty dishes "for Mum", so they'd all just bring them out. THAT, though, was when I often had to "un-teach" people that sometimes (especially holidays) I preferred that everyone just leave their dish, so I could "factor it into" the overall dish-clean-up effort and system I have in place (as that "one person in charge of the clean-up effort"). Somewhere along the way they just grew up to be people who wash their own dishes unless they're asked not to. :)
No "formal" teaching, though.