As Professor Golann details, the mediator gains all of her power from controlling the process of the negotiations, but the advocates can in fact achieve a better outcome for their clients if they are successful in getting the mediator to help them, for example, to make sure that all necessary parties are at the negotiating table, that a joint session is either intentionally held or omitted, that the mediator avoid "reactive devaluation" by delivering key messages instead of delivering them yourself directly to your adversary, and so on.
The integrity of the process is sacrosanct. As Golann concludes, "mediation is an active process, capable of almost infinite variation. Good lawyers know a mediator can help them negotiate effectively, and are not bashful about asking for help." Just be careful to maintain your balance!