Agree with this and even with the point that the OP made about the child knowing what is going on. That may be the case, but still likely the mother is to blame and being the mother, the child is learning the gossiping ways of her mother even if not entirely being pushed by the mother telling her to "spy". I did not break free from the dysfunctional dynamics until into my 20s and even further did not see the extent of it for possibly 10 more years. You play a role as a child sometimes willingly yet ignorantly and sometimes coerced into things but either way the child many times does not have much freedom in acting on their own, or at least does not believe they have that freedom.

I know this makes it no less annoying and disconcerting though. Just kindly keep telling her that it's none of her business.