To download private messages from PF, users can select the desired messages, check the boxes on the right, and choose the "Save" option from the menu at the bottom. Messages can be saved in various formats, including text. Alternatively, users can create a new box to transfer messages, though the capacity is unspecified. There is also a delete option, but caution is advised as deleted messages cannot be recovered. This process helps in managing and clearing out one's mailbox effectively.