I don't know, bitch is pretty common as in most everybody says it or gets called it at some point, but I've never actually heard anybody get called a "dog" before.
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A woman being called a 'bitch' versus a 'dog' has a completely different connotation. Bitch usually denotes strong displeasure. Dog, on the other hand, suggests a certain level of inferiority between the speaker and the target. I'd probably find being called a dog more offensive. If someone calls me a bitch it's probably just because they're angry at me.
"Bitch" is much more of a common derogatory then "dog" is. Using the word "dog" sounds rather silly if you ask me. And both shouldn't be used at all. But when it comes down to it, "bitch" sounds less worse.
It really depends on what the meaning/intend behind the words are though. I could imagne that the word "dog" might be used as a compliment, referring to the positive aspects (cute, cuddly, friendly, loyal, comradeship) but in most instances being meant to express the "superior status/position" of the speaker over the subject.
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