I also agree that when we rescue a dog from unhappy circumstances, they DO know and understand that we have saved them, and they are grateful.
My Gladiator Doberman, Bruno, was a rescue dog. Bruno was a purebred Dobie, and he had been professionally trained as a guard dog. When the local mill went on strike, they purchased Bruno to guard the mill so that the striking workers would not come in and destroy mill property.
After the strike was over, the mill was going to have Bruno put to sleep, but one of the mill workers asked to take him, and he was given to that worker.
It was a year or so later that I came upon Bruno. I was working my insurance sales route, and Bruno was tied out in a muddy yard on a long heavy chain. He was freezing in the cold early spring weather, his food was scattered in the mud and the birds were pecking at it. He hated that, but he was too cold and skinny to even bother chasing them away. You could see his backbone sticking out, and every rib was showing.
I asked the people if they would give Bruno to me, and they agreed. I loaded him into the covered back part of my pickup, took him along with me , and put a big pan of food in the center of my spare tire laying in the back. Some warm blankets for Bruno to lie on, and he was a happy dog again !
I had him for many years, and I know that he appreciated me for taking him home that cold wintery day.