Networks deployed in contested environments typically have limited resources and face many security threads including information leakage. Multicast traffic engineering with carefully considering eavesdropping attacks can improve network performance while avoiding the information leakage problem. In this paper, we address the problem of achieving multicast traffic engineering based on link-state routing protocols according to the offered traffic while taking the problem of alleviating eavesdropping attacks into consideration. We first provide an Integer Linear Programming (ILP) formulation that finds the optimal link weights for Shortest Path Tree (SPT)-based multicast routing protocols to minimize total network cost. The problem is NP-Hard, and obtaining a solution to the problem in real-time to adapt to highly dynamic traffic demands is very challenging. To meet the real-time requirement, we design a Multi-Agent Reinforcement Learning (MARL) solution to the problem of optimizing link weights to achieve efficient multicast routing in a distributed fashion. In our design, the agents collaborate and communicate with the others in the local region and learn from their experiences to determine the best action to minimize the overall network cost. Our proposed solution is evaluated on a simulation of various traffic profiles and compared with the conventional manually configured link weights and a Genetic Algorithm (GA)-based heuristic solution. Experimental results show the advantages of our solutions in reducing network cost and suggest the potential of using MARL in achieving efficient multicast traffic engineering.