Mission-oriented drone networks have been widely used for structural inspection, disaster monitoring, border surveillance, etc. Due to the limited battery capacity of drones, mission execution strategy impacts network performance and mission completion. However, collaborative execution is a challenging problem for drones in such a dynamic environment as it also involves efficient trajectory design. We leverage multi-agent reinforcement learning (MARL) to manage the challenge in this study, letting each drone learn to collaboratively execute tasks and plan trajectories based on its current status and environment. Simulation results show that the proposed collaborative execution model can successfully complete the mission at least 80% of the time, regardless of task locations and lengths, and can even achieve a 100% success rate when the task density is not way too sparse. To the best of our knowledge, our work is one of the pioneer studies on leveraging MARL on collaborative execution for mission-oriented drone networks; the unique value of this work lies in drone battery level driving our model design.