Without treading into the dangerous waters of creating or highlighting any conflicts of interests between lawyer and client, a good mediator will ask a lot of rhetorical questions and be a leader in this regard, based upon the claims being made by each side and her independent evaluation of what is truly needed in order to resolve the dispute. The key is transparency so that neither side feels that you are working on their behalf, while underscoring the most likely outcomes if the matter is not settled, including the risks that one or the other side is correct in their new evaluation!
Last week, I mediated a small case in which not only did the defense lawyer think his clients should pay the Plaintiff's bill, but one of the two partners believed it would be more expedient and less expensive to pay than to continue to fight. The other, of course, in a dramatic fashion, adamantly maintained that this was extortion and he would rather lose the case in trial than pay for a claim which he did not believe had merit. There were lots of micro-opportunities to mediate in only the defense room!
This kind of mediation is a great challenge. A fearless mediator is one that dives into these micro-conflicts before they break out into nasty accusations that potentially derail the negotiation at hand.