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Deep Reinforcement Learning (DRL) has shown its capability to solve the high degrees of freedom in control and the complex interaction with the object in the multi-finger dexterous in-hand manipulation tasks. Current DRL approaches lack behavior constraints during the learning process, leading to aggressive and unstable policies that are insufficient for safety-critical in-hand manipulation tasks. The centralized learning strategy also limits the flexibility to fine-tune each robot finger's behavior. This work proposes the Finger-specific Multi-agent Shadow Critic Consensus (FMSC) method, which models the in-hand manipulation as a multi-agent collaboration task where each finger is an individual agent and trains the policies for the fingers to achieve a consensus across the critic networks through the Information Sharing (IS) across the neighboring agents and finger-specific stable manipulation objectives based on the state-action occupancy measure, a general utility of DRL that is approximated during the learning process. The methods are evaluated in two in-hand manipulation tasks on the Shadow Hand. The results show that FMSC+IS converges faster in training, achieving a comparable success rate and much better manipulation stability than conventional DRL methods.