The short answer to your question is that Calpurnia is the one who tells Atticus about the dog. Scout and Jem see the dog acting all funny. They run and tell Calpurnia. She calls Atticus on the telephone as soon as she sees the dog for herself.
The reason that she she calls is because she believes that the dog has gone mad. This would make it a danger to anyone who is outside. Atticus comes home with the sheriff. Atticus shoots the dog after the sheriff asks him to do it.
In chapter 10 the children see the dog coming down the street falling sideways like he is drunk. The dog is going slow and acting strange. They run to Calpurnia and tell her.
Calputrnia calls Atticus and also tells the operator to call the people and tell them to shut their doors. She even runs to Boo Radley's house to warn them.
Atticus shows up and demonstrates his abilities as a crack shot. The children are amazed as they did not think that their father could do anything. They think of him as being old. He kills the dog with one shot.