Con artists and other Personalities are played when you contribute to a Problem that is successfully solved (i.e. the Problem card’s resource cost is paid and the Problem is discarded). These Personalities may be played on a deal you close (when a Problem is solved on your turn), or on a deal you help to close (when a Problem is solved on another player’s turn).
Some Personalities with this trigger are also Sly, meaning the card can be played anytime up to Step 3 of closing a deal.