There’s no need to have the other user go through a new Bluetooth pairing procedure since a pet profile can be “shared”.
From the FitBark mobile app:
Tap on your pet's profile picture (top right)
In the PET SETTINGS menu, select "Owners & Caregivers"
Tap on the "ADD HUMAN" button
Insert the email address of the user whom you’d like to invite
Select whether the user you are inviting is an owner or caregiver
For caregivers, select whether to share your pet's hourly activity, and/or location
Select “SEND INVITE” to send the email invite