Just using an objective intermediary can produce settlements that parties can't reach in direct confrontation. That's why mediation is a popular tool for resolving conflicts.
Using lawyers as mediators has some added benefits:
- Lawyers know the law and rules of litigation better than laymen and so they can tell their clients what the range of possible outcomes are, and when communicating with opposing counsel they both know what is realistic.
- Lawyers are (at least in principle) forbidden to lie in the course of their professional conduct.
I.e., using lawyers should reduce the amount of gaming involved in resolving a conflict.
And if that's not reason enough to get opposing lawyers speaking directly: Rules of litigation require opposing counsel to "meet and confer" regarding many procedural issues.
A typical example is CCP §2023.010 which requires the court to sanction a party for "Failing to confer in person, by telephone, or by letter with an opposing party or attorney in a reasonable and good faith attempt to resolve informally any dispute concerning discovery."
The judicial system thinks it's always a good idea for opposing lawyers to speak directly because it can only reduce the burden on the public courts. Lawyers are officers of the courts to which they have been admitted to practice, and so while their primary duty is to their clients, they also have an obligation to uphold the law and avoid misusing the courts. Here's an essay published by the ABA that further explains, "cooperation represents the shortest, fastest, and least costly path to what the rules, as applied, ultimately would require the parties to do anyway."
From the perspective of the client, the only drawback to conference that I can think of is the theoretical risk that one lawyer will unnecessarily divulge some piece of information that the other party can exploit to its advantage. Such a mistake would be grounds for a malpractice claim against the lawyer who committed it. And if it is brought to the attention of a court overseeing the process then the court may take steps to mitigate the damage to the party so disadvantaged. A common example is in criminal law where a judge will prohibit improper statements or evidence from being exposed to the jury.