Network utility maximization (NUM) is a general framework for optimally allocating constrained resources in many networked applications. When agents have asymmetric and private information, a fundamental economic challenge is how to solve the NUM Problem considering the self-interests of strategic agents. Many previous related works have proposed economic mechanisms that can cope with agents' private utilities. However, the related literature largely neglected the issue of information asymmetries regarding constraints, and limited closely related studies provided solutions only applicable to specific application scenarios. To tackle this issue, we propose the DeNUM Mechanism, the first mechanism for solving a general class of decomposable NUM Problems considering both private utility and constraint information. The key idea is to decentralize the decision process to agents, who will make resource allocation decisions without the need of revealing private information to others. We further show that the DeNUM mechanism yields the network-utility maximizing solution at an equilibrium, and achieves other desirable economic properties (such as individual rationality and budget balance). However, the corresponding equilibrium solution concept, the generalized Nash equilibrium (GNE), makes it difficult to achieve through a distributed algorithm. To address this issue, we further establish the connection between the structure of GNE and that of the primal-dual solution to a reformulated NUM problem, based on which we present the convergent DeNUM Algorithm that is provably convergent. Finally, as a case study, we apply the DeNUM Mechanism to solving the NUM problem for a user-provided network, and show that the DeNUM algorithm improves the network utility by 17% compared to a non-cooperation benchmark.