Understanding the flaw in a proof attempt of the Communication Complexity of Equality
This vague argument can be made to work for deterministic communication complexity. In the randomized setting, protocols are allowed to err. Consider the standard protocol in which both players exchange a random hash of their inputs to some set of size $2^m$. They only learn $m$ bits of each other's input. When both hashes agree, they aren't completely sure that $X=Y$, and in particular no party has learned the other party's input.