After the Meet & Greet and return discussion, if both you and the private owner are ready to finalize the adoption, the owner will log in to their Rehome by Adopt a Pet account and click "Finalize the Adoption" on your application. This will automatically trigger an email to both of you with a Transfer of Ownership form.
For more info on the Rehome adoption process, click here and start with #5 for next steps after a Meet & Greet!