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