My roommate from the Congo said when he lived there, his neighbor regularly hung dogs by their necks from a rope on his porch and then beat them until most of their bones were broken before cooking them. I'm assuming they were half-tame dogs who would approach humans for food.
"Cat-In-A-Bag" is also a meal that I've heard of, where one puts a cat in a burlap sack, hangs it from a rope on the porch and then beats it to death with a bat and then cooks it.
At the end of the CNN article about the terrier who was BBQ'd, it says the City Mayor will not seek to illegalize eating pet dogs, but will seek to educate new immigrants of New Zealand's customs.
From an objective viewpoint, it takes a special "mentality" to eat a pet. Not to kill it, but to nonchalantly dine on its body for a Saturday cookout. The Mayor is overlooking a more serious problem. Child Protective Agencies know that when there is animal abuse in a home, there is a 95% chance of child abuse as well. By law, Child Protective Services are required to investigate any home where animal abuse occurs.
The type of man who can BBQ a pet has a high risk of also nonchalantly committing some heinous child abuse too, but no one seems to be mentioning that.