Cancer progression is driven by cell proliferation, apoptosis, and matrix invasion, which in turn depend on a myriad of factors including microenvironment stiffness, nutrient supply, and intercellular communication. Cell proliferation is regulated by volume, but in 3D clusters it remains unclear how multiple cells interact to control their size. In this study, we propose a mechano-osmotic model to investigate the evolution of volume dynamics within multicellular systems. Volume control depends on an interplay between multiple cellular constituents, including gap junctions, mechanosensitive ion channels, energy consuming ion transporters, and the actomyosin cortex, that coordinate to manipulate cellular osmolarity. In connected cells, mechanical loading is shown to significantly affect how these components cooperate to transport ions, and precise volume control is impacted by the emergence of osmotic pressure gradients between cells. Consequent increases in cellular ion concentrations drive swelling, while a loss of ions impedes the compression resistance of cells. Combining the modeling framework with novel experiments, we identify how gap junctions can amplify spatial variations in cell volume within multicellular spheroids and, further, describe how the process depends on proliferation-induced solid stress. Our model provides new insight into the role of gap junctions in cancer progression and can help guide the development of therapeutics that target inter- and extra-cellular ion transport. \#\#\# Competing Interest Statement The authors have declared no competing interest.