Boy, if you do that at my house, the garbage man won't take it. They dump it out and there will be a mess in front of your house. Yard waste has to be put in paid-for municipal bags. And if you cheat and put it in regular bags, the garbage guys will know it and just leave the bag laying in front of your house.
I would not accept this. I would go over or call her on the phone and ask her to please not use your trash can without asking, and to never put yard waste in it.
If you have a garage, I would keep the garbage can in the garage until right before pickup, if possible.
Am I stingy about my can? No. But if I can't get my garbage in my can because of my neighbor, it's a problem. It certainly is a big problem if the garbage man dumps things because they aren't supposed to be there.
ETA - What you wrote about the high-end neighborhood you live in and your neighbors being people with means doesn't bother me one bit. I agree with you that people who have the money to pay for their own services have no business stealing yours. And yes, that actually is what they are doing. If you are neighborly with them, then you don't want to hurt your relationship, but come on! You can't get your garbage in your own can because they are using your can as their own! No way!