In a hybrid deployment of on-premises Microsoft Exchange Server and Microsoft Exchange Online in Microsoft Office 365, you can configure free/busy information sharing so that on-premises users and cloud-based users can see each other's availability in Microsoft Outlook or Outlook Web App.
However, you can't grant permissions to create or edit items in another user's calendar cross-premises. That is, cloud-based users can't edit the calendars of on-premises users, and on-premises users can't edit the calendars of cloud-based users. For this level of access, especially when one user must act as a delegate for another user, both mailboxes must exist in your on-premises Exchange organization, or both mailboxes must exist in your Office 365 organization.
To move an on-premises mailbox to the cloud-based organization, or to move a cloud-based mailbox back to the on-premises organization, run the New Remote Move Request Wizard in the Exchange Management Console on the hybrid server. For more information about mailbox migration between the on-premises organization and the cloud-based organization, see the following wiki post in the Office 365 Community:
Error messages that users may receive when they try to grant Write-level permissions or perform Write-level actions (such as deleting an appointment in another user's calendar in Outlook or Outlook Web App) include the following:
One or more users cannot be added to the folder access list. Non-local users cannot be given rights on this server
You cannot make changes to the contents of this read-only folder
You don't have the appropriate permission to perform this operation
Note Calendars are shared across organizations are read-only. The owners of these calendars aren't added to a meeting invitation when you click New Meeting with All in the Outlook calendar. To fix this, manually add the owners to the meeting invitation.