We explore the fundamental limits of distributed balls-into-bins algorithms. We present an adaptive symmetric algorithm that achieves a bin load of two in log* n+O(1) communication rounds using O(n) messages in total. Larger bin loads can be traded in for smaller time complexities. We prove a matching lower bound of (1-o(1))log* n on the time complexity of symmetric algorithms that guarantee small bin loads at an asymptotically optimal message complexity of O(n). For each assumption of the lower bound, we provide an algorithm violating it, in turn achieving a constant maximum bin load in constant time. As an application, we consider the following problem. Given a fully connected graph of n nodes, where each node needs to send and receive up to n messages, and in each round each node may send one message over each link, deliver all messages as quickly as possible to their destinations. We give a simple and robust algorithm of time complexity O(log* n) for this task and provide a generalization to the case where all nodes initially hold arbitrary sets of messages. A less practical algorithm terminates within asymptotically optimal O(1) rounds. All these bounds hold with high probability.