The fundamental components of many signalling pathways are common to all cells. However, stimulating or perturbing the intracellular network often causes distinct phenotypes that are specific to a given cell type. This 'cell specificity' presents a challenge in understanding how intracellular networks regulate cell behaviour and an obstacle to developing drugs that treat signalling dysfunctions. Here we apply a systems-modelling approach to investigate how cell-specific signalling events are integrated through effector proteins to cause cell-specific outcomes. We focus on the synergy between tumour necrosis factor and an adenoviral vector as a therapeutically relevant stimulus that induces cell-specific responses. By constructing models that estimate how kinase-signalling events are processed into phenotypes through effector substrates, we find that accurate predictions of cell specificity are possible when different cell types share a common 'effector-processing' mechanism. Partial-least-squares regression models based on common effector processing accurately predict cell-specific apoptosis, chemokine release, gene induction, and drug sensitivity across divergent epithelial cell lines. We conclude that cell specificity originates from the differential activation of kinases and other upstream transducers, which together enable different cell types to use common effectors to generate diverse outcomes. The common processing of network signals by downstream effectors points towards an important cell biological principle, which can be applied to the understanding of cell-specific responses to targeted drug therapies.