Suppose there is only one sick dog. On the first day, after reading it, if someone sees that other people's dogs are all good dogs, then he will know that his dog is sick and shoot. Because there was no gunshot, the assumption is not valid, which means that there are at least two sick dogs, and everyone knows this information.
Suppose there are only two sick dogs, everyone can see them except the owner, and both of them only look at one sick dog. From the information on the first day, they all knew that their dogs were sick and would be shot, but no one would shoot the next day. If the hypothesis is not established, it means that there are at least three sick dogs, and everyone knows this information.
Suppose there are only three sick dogs, and the owners of the three sick dogs see only two sick dogs. From the information on the first day, they all know that their dogs are sick and will shoot their dogs, which is consistent with the problem.
However, this problem is not over. It should be assumed that there are more than three sick dogs. On the third day, the owner of the sick dog saw three or more sick dogs and could not judge whether his dog was sick or not from the information of the previous day. There will be no gunshots at this time, which is inconsistent with the meaning of the question and the assumption is not established.
So there are three sick dogs in the village.