I guess I don't consider a dog having dignity. Yes, they may feel worried when they potty in the house, and we have to just clean it up, not saying it is ok, not punishing the dog, just taking it as a matter of course. But I do not know that, that would be the deciding factor in when to euthanize.
For me it is a quality of life thing. If the dog is in pain, and is depressed -- does not want to get up, does not want to eat, etc. It is probably time. We can help dogs go to the bridge, so they do not need to suffer through the process of dying.
I wonder also if dignity means different things to different dogs? For instance, I know many older dogs are completely mortified by having an accident. I posted a while back about Heidi "losing" poop in the house occasionally. When this happened, she would just look at me like "so what, deal with it." I'm not saying it wouldn't be a dignity issue if she gets to where she is lying in it, etc., but a small, firm accident really didn't seem to bother her much. Also, I know some dogs get along amazingly well using a cart, but I think for others it would signify a loss of dignity?
I am not convinced that dogs understand the concept of dignity. Going potty in the house due to old age/health issues can certainly upset the humans in the house. Dogs read those upset emotions and react to it. Owners that are sensitive to such events react more strongly and so does the dog (e.g. hiding, hanging head, etc which I think is appeasement behavior and not shame or loss of "dignity").
When my old dog Maddie started having incontinence issues we simply dealt with the cleanup, didn't scold or otherwise react to it. It happened. We cleaned and moved on with our lives. And so did Maddie.
Maddie also used a cart and it bought her another year of quality life. Was she ashamed and feeling less dignified by using a doggie wheelchair? No, I don't think so. She took to it immediately and was really happy having increased mobility. She did feel a little vulnerable being locked into an apparatus but I don't think she was ashamed of it. I know plenty of humans, my mother included, that could not accept a wheelchair due to a loss of dignity. If my mother could have only seen Maddie enjoying her wheelchair perhaps it would have convinced her to accept one herself. But I digress...
Getting old sucks. Someday someone may have to clean up after me and I sure hope that isn't enough to have me put down. As Selzer said, it's a quality of life thing. When Maddie didn't want to do her daily walk anymore and seemed depressed we knew it was time to PTS. When I no longer take joy from life, well then my time will have come.