The commodification of computing, sensors, actuators, data storage and algorithms has unleashed a new wave of automation throughout society. Motivated by the promise of new capabilities, quality improvements, or efficiency gains, data-driven technologies have captured the attention and imagination of the public and many domain experts. Though opportunities are ample, the rapid introduction of data-driven functionality also triggers well-founded concerns about safeguarding critical values, such as safety, privacy and justice. In the context of operating electric distribution networks, the need for data-driven monitoring and control is explained by the irreversible transition from fossil to renewable generation and the accompanied electrification of our economy in areas like transportation and heating.The traditional fit-and-forget paradigm of designing networks conservatively for the projected peak loads assumed unidirectional power flow, predictable future demand and monotonic voltage drops, and allowed for operating at near-100\% reliability with minimal requirement for sensing and actuation. The intermittent nature of Distributed Generation (DG), its ability to feed power back to the grid and cause bidirectional power flow, and the diversifying and nonlinear behavior of electric loads are all eating away at the robustness of this approach, causing Distribution System Operators (DSOs) to put caps on the allowable DG and revisit their design and operating practice. Rather than making traditional expensive network reinforcements in often aging physical infrastructures, DSOs are trying to increase the observability and controllability of their networks by leveraging new sensing and actuation technologies and exploring the ability to use data-driven algorithms to help with the integration of more DG in a more distributed (in space and time) and cost-effective way. This dissertation works towards this vision by formulating a systematic control-theoretic approach for integrating data-driven monitoring and control in the operation of electric distribution networks. Firstly, a Bayesian approach to state estimation overcomes the constraint of limited available real-time sensors by integrating voltage forecasting. A second class of tools discussed is the use of machine learning to decentralize Optimal Power Flow (OPF) methods, by utilizing inverter-interfaced Distributed Energy Resources (DERs). The Decentralized OPF method lets each DER learn a policy that contributes to network objectives from its local historical data and measurements alone. This approach is formulated as a compression and reconstruction problem through an information-theoretic lens, providing fundamental limits of reconstruction and a strategy for optimal communication to improve learning-based reconstruction of optimal policies throughout a network. Lastly, the ambition to control networks in a distributed fashion triggers concerns about privacy-sensitive information that may be inferred from an agent's shared data. For a general class of algorithms, a new notion of local differential privacy is integrated that allows each agent to customize the protection of local information captured in constraints and objective functions. The ultimate goal of the work presented in this dissertation is to contribute to a framework for the integral and value-sensitive design and implementation of data-driven methodologies in critical infrastructure. To address the inherent cross-disciplinary nature of this larger goal, the final chapter explains how each automated decision-making tool reflects and affects values important to its stakeholders. The chapter argues that in order to enable beneficial integration of such tools, practitioners need to reflect on their epistemology and situate the design of automated decision-making in its inherently dynamic and human context.