Give each person 1 - 3 random pieces of the puzzle at the beginning of a gathering/meeting. Have them write their name on the backside of each piece.
During a break or other mingling time, people try to find others whose pieces fit with theirs. This should elicit conversations between people who don't necessarily know each other. When there is a pairing, put those pieces on a table where the whole puzzle eventually gets assembled.
The pieces can be somehow affixed to nametags or displayed by themselves on people's shoulders, as long as they are still removable and all sides available to see if they fit with others.
The organizer can start with a blank jigsaw puzzle (available in some art, toy, and craft stores), and write a motto or picture related to the team on one side.