In the electrical grid, operations such as averaging of local power profiles must be executed without relying on a central aggregator node. It is important to do this in a privacy-preserving way, meaning that devices do not reveal their sensitive information to others. This paper presents a method for devices in a distributed network to approximate the average state of devices without revealing their sensitive local power profiles to each other. We use a distributed load-balancing averaging algorithm from literature and enhance its privacy by locally injecting random zero mean Gaussian noise while providing a method to keep the noise in the noisy network average under a pre-determined tolerance bound Φ. We show that the added noise preserves nodal privacy but cancels out as the network size increases, preserving the practical utility of the network average. For example, in networks with 500 and 1000 devices, and a predefined permissible deviation threshold of 60% between the network average and the true average, our approach keeps the noisy network average within a deviation of just 4.32% and 2.92% from the true converged state respectively.