"We're not" vs. "we aren't"

I'm just curious if there are any "official" rules (or opinions either way) about what form to use when three words can be contracted on either side.

Well, since they do mean the same thing, I would think that they are generally interchangeable. Both are grammatically valid, but there is one reason to use one over the other; if you are trying to increase the emphasis on one word, don't contract it. For example, if I am trying to emphasize the fact that we're not [something]:

We're not criminals.

If you are trying to emphasize the fact that there are multiple people, you might use we aren't:

We aren't going to let you take our pets!

(To me, the first has more emphasis on not than the second does on we.)

That's the only difference I can see; other than that, there probably aren't many others. Also, NGrams claims that we're not is more popular: