That decision belongs to both you (as the potential adopter) and the private owner.
If you both want the pet to go to their new home right after the Meet & Greet, the private owner will need to be able to finalize the adoption (online through Rehome by Adopt a Pet) while at your public Meet & Greet location.
For more tips on this and setting your new pet up for success in your home, check out our step-by-step guide to adopting a pet through Rehome, starting with #5!