Coordinated actions of cells in microbial communities and multicellular organisms enable them to perform complex tasks otherwise difficult for single cells. This has inspired biological engineers to build cellular consortia for larger circuits with improved functionalities, while implementing communication systems for coordination among cells. Here, we investigate the signalling dynamics of a phage-mediated synthetic DNA messaging system, and couple it with CRISPR interference to build distributed circuits that perform logic gate operations in multicellular bacterial consortia. We find that growth phases of both sender and receiver cells, as well as resource competition between them, shape communication outcomes. Leveraging the easy programmability of DNA messages, we build 8 orthogonal signals and demonstrate that intercellular CRISPRi (i-CRISPRi) regulates gene expression across cells. Finally, we multiplex the i-CRISPRi system to implement several multicellular logic gates that involve up to 7 cells and take up to 3 inputs simultaneously, with single- and dual-rail encoding: NOT, YES, AND, and AND-AND-NOT. The communication system developed here lays the groundwork for implementing complex biological circuits in engineered bacterial communities, using phage signals for communication.