Based on recent works on asynchronous versions of the distributed Alternating Direction Method of Multipliers (ADMM) algorithm, we develop and prove the convergence of a distributed asynchronous method for Production-Sharing Problems over networks. The asynchronous nature of the algorithm allows both for the relaxation of the synchronization constraint often inherent to distributed ADMM-based methods and distributed optimization methods at large, but also allows for random local failures to occur in fully centralized methods. These two considerations motivate the application of the method to the Direct-Current Optimal Power Flow (DC-OPF) problem in power transmission networks. Applied to the DC-OPF, this method leads to an overall network optimal production obtained through a sequence of local computations in subareas of the network (each area waking up randomly while the rest of the network is non-operational) and neighboring data exchanges. In another scenario, the DC-OPF is performed via iterations of a centralized network-wide ADMM method which may contain disconnected nodes (in general with low probability and for a short duration). In both cases, this method still converges and thus provides additional flexibility to classical DC-OPF algorithms. The proposed algorithm, inherently designed for networks of overlapping subareas, is then extended to networks of nonoverlapping areas. Simulations are carried out on the IEEE-30 and IEEE-118 bus test systems which illustrate the convergence, scalability and effectiveness of the proposed algorithms.