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I wouldn't worry about the diapers, other than the habit of getting into the garbage and the nasty breath. Dogs just aren't as susceptible to those sorts of diseases. If you think about it, dogs are scavengers. They're built to digest all sorts of carrion and other disgusting stuff, and baby waste doesn't comparatively carry much in the way of disease anyway.
It's gross, but the thing you need to worry about is just generally getting into the trash, especially if it's the neighbors. You never know when someone is going to get tired of picking up the mess and just put poison in something for your dog to eat.