We study the problem of finding an approximate maximum matching in two closely related computational models, namely, the dynamic graph streaming model and the simultaneous multi-party communication model. In the dynamic graph streaming model, the input graph is revealed as a stream of edge insertions and deletions, and the goal is to design a small space algorithm to approximate the maximum matching. In the simultaneous model, the input graph is partitioned across k players, and the goal is to design a protocol where the k players simultaneously send a small-size message to a coordinator, and the coordinator computes an approximate matching. Dynamic graph streams. We resolve the space complexity of single-pass turnstile streaming algorithms for approximating matchings by showing that for any ϵ > 0, O(n2-3ϵ) space is both sufficient and necessary (up to polylogarithmic factors) to compute an ne-approximate matching; here n denotes the number of vertices in the input graph. The simultaneous communication model. Our results for dynamic graph streams also resolve the (per-player) simultaneous communication complexity for approximating matchings in the edge partition model. For the vertex partition model, we design new randomized and deterministic protocols for k players to achieve an a-approximation. Specifically, for a ≥ √k, we provide a randomized protocol with total communication of 0(nk/a2) and a deterministic protocol with total communication of 0(nk/a). Both these bounds are tight. Our work generalizes the results established by Dobzinski et al. (STOC 2014) for the special case of k = n. Finally, for the case of a = o(y/k), we establish a new lower bound on the simultaneous communication complexity which is super-linear in n.