Just as a great party has a great host, the problem solving party needs to someone guide the party itself: a facilitator. The facilitator plans how the party will flow, who shows up, how the environment of the party is set up, and helps everyone to have fun.
The responsibility of the facilitator in these processes is to ensure that everyone keeps to the objectives set for the party (remember this party has a purpose), tries to help everyone get involved, and stays out of the content of the problem. (A BIG SECRET)
At the end, the facilitator debriefs the process and figures out where to go next with the owner of the problem. Sounds easy? Depends on your personality, your preferences, and other factors. It could be. In any case, it is a process of being the great party host.
The best place to learn how to be a great facilitator is at the Creative Studies program at Buffalo State College where they offer a certificate for Creativity and Change Leadership.
