Neural circuits comprise multiple interconnected regions, each with complex dynamics. The interplay between local and global activity is thought to underlie computational flexibility, yet the structure of multiregion neural activity and its origins in synaptic connectivity remain poorly understood. We investigate recurrent neural networks with multiple regions, each containing neurons with random and structured connections. Inspired by experimental evidence of communication subspaces, we use low-rank connectivity between regions to enable selective activity routing. These networks exhibit high-dimensional fluctuations within regions and low-dimensional signal transmission between them. Using dynamical mean-field theory, with cross-region currents as order parameters, we show that regions act as both generators and transmitters of activity -- roles that are often in tension. Taming within-region activity can be crucial for effective signal routing. Unlike previous models that suppressed neural activity to control signal flow, our model achieves routing by exciting different high-dimensional activity patterns through connectivity structure and nonlinear dynamics. Our analysis offers insights into multiregion neural data and trained neural networks.