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Answer by Jonas Mikka Luster, restaurant owner, certified master chef:
Some do, some don't. That's a question of restaurant policy and restaurant culture.
There are sendbacks. Those are just eaten by the least well-treated of staff. No one who works in food should ever have to be so hungry that he has to eat someone else's food off a half-eaten plate. Sadly, it happens, especially in chain dining.
Then there are the “there's more in this pan than I need” leftovers. They're eaten when they cannot be reused—they're a loss for the place anyway. It'd be a shame, a disservice, and a disgrace to the animals that died for this food or workers who labored for it to throw it away. Stuff like that doesn't make a full meal, though.
End-of-shift leftovers are another matter. They're a direct loss for the kitchen. Might as well eat them, yes, but they're also a sign that your chef sucks. A good kitchen has very little leftovers at the end of the night.
Soon-to-expires do happen, even in very good places but much less than in not-so-good ones. There's no reason not to eat what can't be sold and would go bad otherwise. Read above about disservice.
Now for the meat of the matter. No employee of a restaurant should ever have to eat leftovers. We're working in food, so this would be a little bit like a cobbler going barefoot. A good restaurant has “family meals” (cooking for the staff by the staff) and a good policy about taking things for oneself as long as it's not excessive. In my places, I have always insisted that food, coffee, water, and soda are free, as well as one or two alcoholic beverages after shift. I've also worked in a place where our “benefits” included “10 percent off your meals here,” at which point I made it a habit to walk in there with food from a competing restaurant or the nearby McDonald’s. It's just bullcrap to have such a rule.
On the turn, if the staff doesn't want to eat what it cooks or serves, would you want to eat there?