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Image domain prior models have been shown to improve the quality of reconstructed images, especially when data are limited. Pre-processing of raw data, through the implicit or explicit inclusion of data domain priors have separately also shown utility in improving reconstructions. In this work, a principled approach is presented allowing the unified integration of both data and image domain priors for improved image reconstruction. The consensus equilibrium framework is extended to integrate physical sensor models, data models, and image models. In order to achieve this integration, the conventional image variables used in consensus equilibrium are augmented with variables representing data domain quantities. The overall result produces combined estimates of both the data and the reconstructed image that is consistent with the physical models and prior models being utilized. The prior models used in both domains in this work are created using deep neural networks. The superior quality allowed by incorporating both data and image domain prior models is demonstrated for two applications: limited-angle CT and accelerated MRI. The prior data model in both these applications is focused on recovering missing data. Experimental results are presented for a 90 degree limited-angle tomography problem from a real checked-bagged CT dataset and a 4x accelerated MRI problem on a simulated dataset. The new framework is very flexible and can be easily applied to other computational imaging problems with imperfect data.