The owner is able to view your location and can track your movements. You can disable them after they are found or follow the steps here to help identify them. Pressing the button on the AirPods to reset them will not remove the original owner and will remain locked to them. You will not be able to set them up as yours. It is just the Notification that can be suspended for one day or indefinitely when you are part of a Family Sharing group with the owner.
If you're borrowing an AirTag or another item compatible with the Find My network from a member of your Family Sharing group, you can turn off alerts for that item for one day, or indefinitely.
What to do if you get an alert that an AirTag, set of AirPods, Find My network accessory, or compatible Bluetooth location-tracking device is with you - Apple Support