Amphiphilic inducer molecules such as N-acyl-L-homoserine lactones (AHLs) or isopropyl-beta-D-thiogalactopyranoside (IPTG) can be utilized for the implementation of an artificial communication system between groups of E. coil bacteria encapsulated within water-in-oil microemulsion droplets. Using spatially extended arrays of microdroplets, we study the diffusion of both AHL and IPTG from inducer-filled reservoirs into bacteria-containing droplets, and also from droplets with AHL producing sender bacteria into neighboring droplets containing receiver cells. Computational modeling of gene expression dynamics within the droplets suggests a strongly reduced effective diffusion coefficient of the inducers, which markedly affects the spatial communication pattern in the neighborhood of the senders. Engineered bacteria that integrate AHL and IPTG signals with a synthetic AND gate gene circuit are shown to respond only in the presence of both types of sender droplets, which demonstrates the potential of the system for genetically programmed pattern formation and distributed computing.