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An update to this old question:

• _Currently_, MoveIt does not support executing multiple trajectories via a MoveGroupInterface or Commander interface.
• Sending planned trajectories directly to the robot controllers is possible, but MoveIt will not ensure that there are no collisions between the trajectories. This is dangerous and should be done at your own risk.
• It is possible to move multiple arms at the same time, but they both need to be part of one planning group and trajectory. They cannot start and stop their motion independently of the other. When planning, you need to set a goal for each arm (or group of joints). This project has some examples (check the lines in the o2ac_routines package that use the planning group ab_bot).
• In the same project, we also forked MoveIt to make a single move_group move multiple PlanningGroups independently. This is the functionality requested in the original question.
• This open PR should bring the changes into upstream MoveIt and enable multiple trajectory execution. Finishing this big PR is the subject of a Google Summer of Code project, tracked here. It is a big change, so you can support it by reviewing and following the issue and PR.