One of the key skills of an effective mediator is being quickly able to obtain the trust of the parties. Trust that the mediator is independent and impartial, trust that s/he is a safe pair of hands and that they won’t be disadvantaged by showing vulnerability and exposing weaknesses in private session, and so on.
But the deal is two-way because the parties and their advisers will also benefit from earning the trust of the mediator. The mediator can work much more effectively if parties are being frank and open in private session with him/her, where the mediator is involved in the exchanging of information and discussion of offers. If the mediator doesn’t trust the parties, then s/he is going to be far more inhibited in using his/her unique position to get to a deal.