We do know (see also) that mindfulness training can bring a person many benefits including increased emotion regulation, response flexibility, focus, and self-insight; and decreased reactivity, anxiety, and stress. All of those benefits would seem to support the effectiveness of a mediator but we have no direct research on mediator mindfulness or self-awareness—yet.
Until research is done on mediators, we will need to look for clues about factors of efficacy in other professions. At least one program for teachers is showing the benefits of mindfulness training. Go here to read some articles on the program. At that same link is a logic model that shows the influence of the social and emotional competence of the teacher on his or her classrooms and students. Wouldn't it be fun to draw an analogous diagram that shows the effects of mediators' social and emotional competence (I would add cognitive maturity) on the conflict resolution process and the parties? I just may do it; if you create one, let's compare models.