You can add contacts only 1 way within the Queue: by entering an email or mobile number in the TO field and it will automatically save recipients information to the contacts list of all Queue users regardless of who entered the recipients information. However, you will also need to select a Queue before your request is sent.
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Just remember to go back into your contacts list to edit and add a name to your contact. Once YOU make a change to that contact, the changes will be reflective in only YOUR contact list.
Once a contact is in your contacts list, you can type an email, mobile or name into the TO field and it will auto generate from your contacts list in alpha or numeric order to match your entry so you do not have to type the full address etc.
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If your team has P2 Conversations enabled, then all users contacts within YOUR team are merged into the P2 contacts as Team based contacts vs user only contacts. ORG Owners/Admin will be able to access all teams contacts.