The subtext is you'll sue if your opponent declines the offer. Osborne says her ex-partner settled because the mediator showed him that he had plenty to lose if the spat went to court. Then the mediator talks privately with each side. When that's impossible, the parties can go to court. If there's a meeting of the minds, they usually draw up a binding contract, says Lovenheim.
Lawyers get into the act as legal advisers to the parties or mediators.
When the stakes are high, you may want a lawyer at your side. At least, consult with one before you sign the final agreement. Mediators are unlicensed so their training and experience can vary widely. Facilitators often have counseling backgrounds. Evaluators tend to be lawyers.