Flash sale is reselling your ticket to the next buyer of a ticket on the Shotgun platform, which means payment goes through Shotgun, and you are refunded directly to your bank account, less the resale fee.
Private sale is a transfer, that is to say that you transfer your ticket via the platform indicating the information of the person to whom you send it, and you organise amongst each other concerning payment, which does not take place on the Shotgun platform.