Well it depends on what you believe, I think, whether you believe killing means that this person's suffering has ended. I suppose that is a belief, isn't it? A belief that death is the ending of suffering, and that by killing this being you have ended their suffering. But I don't know whether all of us really know that, do we? That is just a belief. How can we be sure? Maybe this doubt can help us to open more to Dukkha, and see suffering as an opportunity for ourselves and perhaps for this being. Each of us, according to the teachings, is the owner of our own Kamma. Why this being is suffering, usually the teachings point out, is because of their own Kamma. I would prefer to give them some relief from their pain if possible, and try to help them mentally, and see it as an opportunity for this person to actually grow spiritually.
I heard a story a long time ago about a woman who was terminally ill, who was in a great deal of pain and suffering, and was very angry. Apparently all the nurses hated to go to see her because she was very angry. Whenever she called, pressed the button, the nurses would be a long time coming. And one by one she alienated a lot of her family as well. Now if we look closely, if a person is in a lot of pain, usually anger is a response if they have no mental tools to deal with it; anger and fear. So, if we can see below the anger and see the need of this woman, perhaps we will have a little bit more patience, and not turn away from the Dukkha and the suffering at that particular time. Well, she apparently had some good fortune that someone was trying to help her with some meditation techniques. Fortunately, she also knew a bit of a universalizing technique, and apparently one night she opened to her Dukkha, and was able to universalize it, and opened to the Dukkha of the human condition, contemplating how all human beings are subject to suffering. That night was very important for her, because she changed her view totally of her situation, and in the morning a lot of her anger was gone. And then she became the favorite patient of the nurses, because she had had a glimpse into suffering and its cause.
For other beings, it is more difficult because we don't like to see the suffering and we may not be able to communicate with them. So, it becomes a little bit more complicated. As I said, how can we know it is the ending of suffering? As far as that goes for wild creatures, perhaps trying to make their last moments a little bit more comfortable may be helpful. For domesticated animals, well that becomes even more complicated. We try to do all that we can to help cure the illness of the being if it is possible. I wouldn't like to have to make that decision, and that is one reason I don't like to have pets. Because the person who is responsible for them has a very difficult decision to make.