There is a little confusion here about secondary poisoning. It is true that most rat poisons on the market contain anti-coagulants (e.g., warfarin, bromethalin) which absolutely will cause secondary poisoning. But there are alternatives that act on being metabolized and rendered inert. Terad3 is a cholecalciferol based "poison" that, when digested, results in the calcification of organs and subsequent organ (usually kidney) failure. If any of you take joint supplements, you might recognize the ingredient because it's commonly prescribed for joint and bone health. Once it is digested though, it can't affect scavengers. The only chance is if the rat had some undigested product in its stomach or if the rat was caching the product somewhere and a critter found the cache.
A second point on this, to back what some others have said, is that leaving food out definitely will make your rodenticide less effective simply because the rat is used to eating that food and will likely just keep eating that instead of the rodenticide. So if you use poison, take the food away at night at the very least and make poison the only food option for the rats.
I have a rat issue right now, but it's mainly my compost keeping them around. Time for an overhaul and relocation/reconstruction of the compost for me.