Scaling neural network models has delivered dramatic quality gains across ML problems. However, this scaling has increased the reliance on efficient distributed training techniques. Accordingly, as with other distributed computing scenarios, it is important to understand how will compute and communication scale relative to one another as models scale and hardware evolves? A careful study which answers this question can better guide the design of future systems which can efficiently train future large models. Accordingly, this work provides a comprehensive multi-axial (algorithmic, empirical, hardware evolution) analysis of compute vs. communication (Comp-vs.-Comm) scaling for future Transformer models on future hardware. First, our algorithmic analysis shows that compute generally enjoys an edge over communication as models scale. However, since memory capacity scales slower than compute, these trends are being stressed. Next, we quantify this edge by empirically studying how Comp-vs.-Comm scales for future models on future hardware. To avoid profiling numerous Transformer models across many setups, we extract execution regions and project costs using operator models. This allows a spectrum (hundreds) of future model/hardware scenarios to be accurately studied ($<$15% error), and reduces profiling costs by 2100$\times$. Our experiments show that communication will be a significant portion (40-75%) of runtime as models and hardware evolve. Moreover, communication which is hidden by overlapped computation in today's models often cannot be hidden in future, larger models. Overall, this work highlights the increasingly large role communication will play as models scale and discusses techniques and upcoming technologies that can help address it.