Additive Manufacturing is changing the way we construct, deliver, and consume objects by allowing customers to quickly build custom objects on-demand and in locations near to them. Additive manufacturing operators need to optimally control the manufacturing process, monitor tasks in real-time and set the prices for their built objects. Moreover, operators need to protect private data related to the prices paid by customers and their purchased objects. Existing works focus on either optimal control, real-time monitoring, price setting, or privacy, missing the advantages of jointly addressing them. Existing works also require vast computational resources to accomplish only one of these tasks. To address these issues, we develop a differentially-private distributed neural network auction that optimally allocates manufacturing resources and sets prices in a way that maximizes the profit of the manufacturer. The auction protects the privacy of the customers’ bids. Moreover, to reduce the computing time, we design a parallel computation algorithm for the neural network that is executed by a cluster of edge computing devices. We evaluate the proposed neural network through extensive simulations. We observe that it can jointly perform the operators’ tasks while maintaining the privacy of the customers. Our simulations also show that the parallel algorithm reduces the execution time.