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This paper investigates the problem of power allocation for distributed estimation via diffusion in wireless sensor networks (WSNs) with simultaneous wireless information and power transfer (SWIPT). We consider a WSN consisting of smart sensor nodes (SNs) and common sensor nodes (CNs), and each SN is capable of performing SWIPT via multi-antenna beamforming to its neighboring (i.e., near-tier) CNs. In each diffusion iteration, all nodes collect measurements and exchange intermediate estimates with their neighbors. We first analyze the effect of each SN's beamforming design and each near-tier CN's harvested power allocation on the steady-state network-wide mean square deviation (MSD) of the diffusion least-mean-squares (LMSs) strategy. Then, we formulate a problem to minimize an upper bound MSD by jointly optimizing the global power allocation weights for each SN to perform beamforming, and the local power allocation proportion for each CN to perform measurement collection. We further show that the formulated non-convex problem is decomposable and propose a gradient-based iterative algorithm to find the optimal solution. In addition, for practical implementation, we propose adaptive online approaches to estimate some parameters required for system optimization. Finally, extensive simulation results demonstrate that with optimal power allocation, our proposed scheme improves the MSD performance significantly, compared to the conventional diffusion LMS strategy without wireless power transfer (WPT).