No, contacts are saved per device. The contact list in your account is synced to your phone's address book. Since we do not separate personal vs. business contacts in Sideline, they will not share across all team members. Therefore, team members would also need to save the contact to their native address book for the name to show in the inbox.
Example: If you have a message thread with a contact you saved as "Allison" in your phone's address book, your teammate will also need to save the contact on their phone's address book as "Allison" in order for the name to show up on their view of the inbox.
If you have any further questions, please submit a request.