For the Data Sync Exchange Providers to read Calendars,Contacts or Tasks from other mailboxes you will need to add Reviewer permission to the mailboxes for the user that is used in Data Sync. Fortunately you can do this with a PowerShell script like this.
Add-MailboxFolderPermission -Identity email@example.com:\Contacts -User Ed@contoso.com -AccessRights Reviewer
-Identity is mailbox and folder your targeting and
–User is the user to grant access.