Farmers can and do shoot dogs running around amongst livestock
I am not sure this is actually true, and has anyone seen this first hand?
The reason I doubt this is quite simply that the dog and the sheep will be fast moving, and the farmer will be using a shotgun, as if he is using a high powered rifle he has to ensure that the fall zone is safe, so he isn't going to legally be able to use it.
Therefore, if a farmer blasts away at any dog chasing sheep, he is running quite a high risk of hitting the sheep as well as the dog. This combined with the fact, that the effective range of a shotgun is around 20 meters, leads me to believe the 'legend' of farmers shooting the dogs as they chase sheep around a field to be made up by people who don't know what they are talking about, and like to preach shit on internet forums.
Just an observation.
What I suspect happens, is they report the dog for chasing sheep, and if they catch you, you end up with a large fine and the bollocking of your life, second time it happens you end up with an even larger fine and the dog is destroyed.