There have been significant efforts to interpret the encoder of Transformer-based encoder-decoder architectures for neural machine translation (NMT); meanwhile, the decoder remains largely unexamined despite its critical role. During translation, the decoder must predict output tokens by considering both the source-language text from the encoder and the target-language prefix produced in previous steps. In this work, we study how Transformer-based decoders leverage information from the source and target languages -- developing a universal probe task to assess how information is propagated through each module of each decoder layer. We perform extensive experiments on three major translation datasets (WMT En-De, En-Fr, and En-Zh). Our analysis provides insight on when and where decoders leverage different sources. Based on these insights, we demonstrate that the residual feed-forward module in each Transformer decoder layer can be dropped with minimal loss of performance -- a significant reduction in computation and number of parameters, and consequently a significant boost to both training and inference speed.