Shove Thy Neighbor (1957)

Things are peaceful in John Doormat's neighborhood until he gets a new neighbor, and the new neighbor comes with a dog that likes to grab and gnaw on John's leg every time he sees him. Doormat buys his own large mutt which chases the neighbor's pest-pet home. When the two men meet to exchange some bitter words, the neighbor's dog clamps on John's leg, and John's dog is chewing on the neighbor's leg.