Most of these businesses in the district are tiny, family-owned businesses, so there's a huge amount of implicit trust there.
But also, these tiny stones really aren't worth much. What you're asking is like Home Depot making sure to account for every single nut & bolt so none are stolen or lost. It would cost far more in time & labor then what you'd get back in return.
One would think diamond merchants would take the time to be equally careful, if not moreso.