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Depends on the investigators. I'd like to imagine the two of them shadowing each other around a corner, only to come nose-to-nose with each other. Discrete inquiry reveals they were hired by the same guy... then they both go after HIM (the Remington Steele style of detective fiction).
OR...
Confrontation leading to a massive slug-fest or shoot-out, and the loser being dragged to the client by the winner (the Bruce Willis style of detective fiction).
OR...
They lose each other almost immediately, only to stumble upon each other in the bathroom (the Columbo style of detective fiction).
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