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Agents in dynamic multi-agent environments must monitor their peers to execute individual and group plans. A key open question is how much monitoring of other agents' states is required to be effective: The Monitoring Selectivity Problem. We investigate this question in the context of detecting failures in teams of cooperating agents, via Socially-Attentive Monitoring, which focuses on monitoring for failures in the social relationships between the agents. We empirically and analytically explore a family of socially-attentive teamwork monitoring algorithms in two dynamic, complex, multi-agent domains, under varying conditions of task distribution and uncertainty. We show that a centralized scheme using a complex algorithm trades correctness for completeness and requires monitoring all teammates. In contrast, a simple distributed teamwork monitoring algorithm results in correct and complete detection of teamwork failures, despite relying on limited, uncertain knowledge, and monitoring only key agents in a team. In addition, we report on the design of a socially-attentive monitoring system and demonstrate its generality in monitoring several coordination relationships, diagnosing detected failures, and both on-line and off-line applications.