This paper studies task-oriented edge networks where multiple edge internet-of-things nodes execute machine learning tasks with the help of powerful deep neural networks (DNNs) at a network cloud. Separate edge nodes (ENs) result in a partially observable system where they can only get partitioned features of the global network states. These local observations need to be forwarded to the cloud via resource-constrained wireless fronthual links. Individual ENs compress their local observations into uplink fronthaul messages using task-oriented encoder DNNs. Then, the cloud carries out a remote inference task by leveraging received signals. Such a distributed topology requests a decentralized training and decentralized execution (DTDE) learning framework for designing edge-cloud cooperative inference rules and their decentralized training strategies. First, we develop fronthaul-cooperative DNN architecture along with proper uplink coordination protocols suitable for wireless fronthaul interconnection. Inspired by the nomographic function, an efficient cloud inference model becomes an integration of a number of shallow DNNs. This modulized architecture brings versatile calculations that are independent of the number of ENs. Next, we present a decentralized training algorithm of separate edge-cloud DNNs over downlink wireless fronthaul channels. An appropriate downlink coordination protocol is proposed, which backpropagates gradient vectors wirelessly from the cloud to the ENs.