Many biological, physical, and social interactions have a particular dependence on where they take place; e.g., in living cells, protein movement between the nucleus and cytoplasm affects cellular responses (i.e., proteins must be present in the nucleus to regulate their target genes). Here we use recent developments from dynamical systems and chemical reaction network theory to identify and characterize the key-role of the spatial organization of eukaryotic cells in cellular information processing. In particular, the existence of distinct compartments plays a pivotal role in whether a system is capable of multistationarity (multiple response states), and is thus directly linked to the amount of information that the signaling molecules can represent in the nucleus. Multistationarity provides a mechanism for switching between different response states in cell signaling systems and enables multiple outcomes for cellular-decision making. We combine different mathematical techniques to provide a heuristic procedure to determine if a system has the capacity for multiple steady states, and find conditions that ensure that multiple steady states cannot occur. Notably, we find that introducing species localization can alter the capacity for multistationarity, and we mathematically demonstrate that shuttling confers flexibility for and greater control of the emergence of an all-or-nothing response of a cell.