Lots of guesses, a little sillyness, and a couple people that have been there and done that.
Having shot quite a few feral dogs and a rail car load of coyotes I can say the most effective rounds have been mostly overlooked. Fast light bullets that deposit energy very quickly are what is needed. Really, think about it. A BIG dog is less than half the size of a deer, and most are less than 60 pounds. Why in the heck would you want a 44, or a 45, or 180 grain bullets in a 357?? Those are not going to start working well until the bullet is well past most dogs, there just isn't enough dog there. Bone structure on a dog is very light, and find an angle that takes more than 6-8" of penetration to get into the heart/lung cavity or the CNS if you can.
Give me a light for caliber HP at warp speed. I want energy deposited ASAP, bullet fragmentation, and massive shock. THAT is what puts a dog down NOW, not later after he has mangled you and he eventually bleeds out from the caliber hole punched clean through him.
The optimum would be a 223, 22-250, 220 swift or something on that line. Rifles excluded I want a 357 magnum with 110's or 125's at warp speed. A 40 with 135's would also work quite well.