At a certain point in the negotiation, when it threatens to stall, but before it hits an impasse, I begin negotiating with each side in hypothetical brackets or ranges until I hear or sense there is a potential overlap. Contrary to the beginning stages of negotiation, this one really becomes a negotiation between mediator and parties, as opposed to party against party.
By the way, there is not always an overlap, so sometimes I have to consider both the "bluff factor" and the "stretch factor". That is, one side may be telling me that the case will never settle below 6 figures ($100,000.00) and I may ultimately discount that by 10-15% and make a mediator's proposal at $85-90,000 on the assumption that the strong assertion of a particular number has some "bluff" to it. The other side may tell me that they will never go above $75,000, but in a mediator's proposal at $85,000, that side will usually "stretch" to get there.
I just read and reviewed Dwight Golann's excellent book, "Sharing a Mediator's Power" where he describes the negotiating dynamic in mediation as a triangle with each opposing party at either side and the mediator as the base of the triangle--negotiating with both and still trying to achieve balance and harmony between the two, standing as their foundation and supporting them in an integral function of her own.
I prefer to think of it as hitting the "sweet spot" where the proposal just abuts the intersection of good ideas and bad ideas to make a sensible means of settling the case after all other efforts through the course of direct and indirect negotiation have failed. And boy do I feel triumphant when I hit the sweet spot and both sides say "yes".
I agree that the mediator's proposal is a good tool in the hand of someone who knows how to use it. Usually within the first two hours of a mediation session I can guess privately what the settlement will be. If the parties can not agree, and neither party asks for a mediator's proposal, and the parties are still amicable, I ask if they want one. If the mediation ends on a bad note and the parties exit in a huff, I wait two days and try to resume negotiations and/or make a mediator's proposal. Many "impasses" end in settlement after heads have had a few days to cool.
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