We prove that any sketching protocol for edit distance achieving a constant approximation requires nearly logarithmic (in the strings' length) communication complexity. This is an exponential improvement over the previous, doubly-logarithmic, lower bound of [AndoniKrauthgamer, FOCS'07]. Our lower bound also applies to the Ulam distance (edit distance over nonrepetitive strings). In this special case, it is polynomially related to the recent upper bound of [AndoniIndyk-Krauthgamer, SODA'09].From a technical perspective, we prove a direct-sum theorem for sketching product metrics that is of independent interest. We show that, for any metric X that requires sketch size which is a sufficiently large constant, sketching the max-product metric d ∞ (X) requires Ω(d) bits. The conclusion, in fact, also holds for arbitrary two-way communication. The proof uses a novel technique for information complexity based on Poincaré inequalities and suggests an intimate connection between non-embeddability, sketching and communication complexity.