Why can animals eat grass that they have pooped on, but humans would get sick?

As far as I am aware, grazing animals like cows or sheep poop in the same meadows where they eat grass, but presumably don't have any problems eating the grass and pooping in the same space. But if humans would eat vegetables that they had pooped on, my understanding is that we would get sick.

Why? Am I incorrect that grazing animals poop where they eat? Are their stomachs more resistant to whatever makes it dangerous?

Thank you!