I always cut the bag in half from top to bottom so that the bag can't get pushed down around the dog's (or other animal's) neck and strangle them. Once the trash hits the dump, wild animals will try to lick out bags, too, and they can get things caught around their necks and strangle themselves when the garbage gets caught on something. I suppose a dog could do that too if it panicked. It only takes a second to cut the bag but it could save lots of heartache.
It's a good idea to cut up bags and plastic wrapping objects even when you don't have a pet in the house. This junk gets into the dump, where wild and stray animals can harmed, too. Those plastic gadgets they use to link beer and soda cans into four- or six-backs -- don't know what they're called, but they're made with four or circular plastic loops. They're evil -- animals get their heads or muzzles or beaks trapped in them. Take your scissors and nip every one of those dratted circles apart.
Wouldn't it be nice if there were some way get rid of the gawdawful overpackaging we have to cope with these days? WHY are two, three, four layers of plastic and cardboard necessary?