Network sharing has become a key feature of various enablers of the next generation network, such as network function virtualization and fog computing architectures. Network utility maximization (NUM) is a general framework for achieving fair, efficient, and cost-effective sharing of constrained network resources. When agents have asymmetric and private information, however, 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 network sharing paradigm introduces the issue of information asymmetries regarding constraints. The related literature largely neglected such an issue; limited closely related studies provided solutions only applicable to specific application scenarios. To tackle these issues, we propose the Decomposable NUM (DeNUM) Mechanism and the Dynamic DeNUM (DyDeNUM) Mechanism, the first mechanisms in the literature for solving NUM Problems considering private utility and constraint information. The key idea of both mechanisms is to decentralize the decision process to agents, who will make resource allocation decisions without the need of revealing private information to others. Under a monitorable influence assumption, 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). We further establish the connection between the equilibrium structure and the primal-dual solution to a related optimization problem, based on which we prove the convergence of the DeNUM Algorithm to an equilibrium. When the agents' influences are not monitorable, we propose the DyDeNUM Mechanism that yields the network-utility maximizing solution at the cost of the balanced budget.