Listen to Whip. I'd call DNR and explain the situation. They will advise you further. They may just tell you to find the owner or they mey tell you to get the local Sheriff involved. Either way, go through the channels before you have some fella with a gun, coming out looking for the guy who shot his dog. Often times, a warning to the owner from law enforcement is all it takes. At least that way, you have something to back you up, should you end up in court.
Let me give an example. Years back, when my wife was young and living with some guy, fresh out of college, they lived on an old farm and were raising chickens. They would come home to find dead chickens all over the yard, and lots of dog tracks. They follwed the tracks and feathers back to the neighbors and saw the dogs by the house. Her boyfriend at the time, stayed home from work one day and actually caught the neighbor dogs (free running, no collar etc.) getting into the chicken house and chasing/killing chickens. He ran them off, then went over and warned the neighbor that he would shoot the dogs if they came over there again. Well they came back, he shot them, then as young, vengeful, and not so bright young men might do, he dumped the dogs in the ditch on his neighbor's farm for him to find. I guess he thought he was sending a message. Well they ended up in court because the neighbor filed charges. The charges regarding shooting the dogs were dismissed, but not without a lecture from the judge. The young fella had to miss work to go to court, so that cost him. Afterwards, there was plenty of bad blood with the neighbor. Enough to cause him to sell his house and move. Take that into consideration before you decide what to do.