Real-time data-driven optimization and control problems over networks, such as in traffic or energy systems, may require sensitive information of participating agents to calculate solutions and decision variables. Adversaries with access to coordination signals may potentially decode information on individual agents and put privacy at risk. We use the Inexact Alternating Minimization Algorithm to instantiate local differential privacy for distributed optimization, addressing situations in which individual agents need to protect their individual data, in the form of optimization parameters, from all other agents and any central authority. This mechanism allows agents to customize their own privacy level based on local needs and parameter sensitivities. The resulting algorithm works across a large family of convex distributed optimization problems. We implement the method on a distributed optimal power flow problem that aims to prevent overload on critical branches in a radial network.@en