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.
If you both need time to think about the match before making a final decision, please take some time (a day or two) to sit on it. If you determine that you want to adopt the pet, you can set up a pet hand off meeting where you can sign the Transfer of Ownership paperwork and pay the adoption fee.
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!