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This work studies distributed primal-dual strategies for adaptation and learning over networks from streaming data. Two first-order methods are considered based on the Arrow-Hurwicz (AH) and augmented Lagrangian (AL) techniques. Several results are revealed in relation to the performance and stability of these strategies when employed over adaptive networks. It is found that these methods have worse steady-state mean-square-error performance than primal methods of the consensus and diffusion type. It is also found that the AH technique can become unstable under a partial observation model, while the other techniques are able to recover the unknown under this scenario. It is further shown that AL techniques are stable over a narrower range of step-sizes than primal strategies.