IME, what happens to bags that are not picked up is that they are moved off the baggage claim, and either moved into/in front of the airlines Lost Baggage office (normally close to the baggage claim) or moved into an area of the airport (normally close to baggage claim) where all bags of this type of left.
At that point, you'd either just go take it, or be required to show baggage claim ticket.
That being said, you do run the risk (however unlikely) of someone just running off with it.