From the manual: "The device will not insert PAID [Privacy asserted ID] in outbound INVITE requests."
I believe the Obi simply substitutes whatever number you specify in your config as the FROM number. This is why many VOIP providers will not allow the call to be processed. If your Obi is registered on the system as user 17771234567 and you try to send a call that says it is FROM 17772345678 they refuse to put the call through.