As someone who has been in both positions, it is certainly always the owners fault. I had a Husky of mine get hit when he found a way out of our backyard. If any of you know Husky's, you know they are smart dogs and love to roam and they find ways out of your yard if you are not careful enough. So I take full responsibility in that situation and I was very grateful the driver called me up from the number on his tags and I was able to get him to the vet and he made a full recovery. That was 10 yrs ago and he just died of natural causes this past year (at the grand old age of 15).
The time I hit a dog, it was a black dog on a dark night and he came rushing onto the street before I even knew that he was near. I was able to stop and only tapped him with the car. He ran off and I went searching for an hour trying to find him. So I am hoping that he turned out all right, because I never know what happened to him that night. I just hope he didn't get hit again. Good rule of thumb, no dog is to be trusted from finding an opportunity to run off, so be careful when having an unleashed dog that is by a road, you never know (squirrel, cat, ball, etc) what could cause them to run off into danger.