BUENOS AIRES -- When celebrity psychoanalyst Gabriel Rolón picked up the phone, the man on the line said he was holding a loaded gun and was ready to blow his brains out.

Mr. Rolón was recently recounting the incident, which took place several years ago, to a rapt audience of his fans here in Argentina, the nation with more psychologists per capita than any other. Mr. Rolón said he told the caller to come right by his office -- but added, "Please don't bring the revolver." When the caller laughed, Mr. Rolón says, he thought...