In this paper, we explore a multi-agent reinforcement learning approach to address the design problem of communication and control strategies for multi-agent cooperative transport. Typical end-to-end deep neural network policies may be insufficient for covering communication and control; these methods cannot decide the timing of communication and can only work with fixed-rate communications. Therefore, our framework exploits event-triggered architecture, namely, a feedback controller that computes the communication input and a triggering mechanism that determines when the input has to be updated again. Such event-triggered control policies are efficiently optimized using a multi-agent deep deterministic policy gradient. We confirmed that our approach could balance the transport performance and communication savings through numerical simulations.