My clarifying comments follow each bullet, in italics.
- On-premises mailbox permissions such as Send As, Receive As, and Full Access that are explicitly applied on the mailbox are migrated to Exchange Online. Inherited (non-explicit) mailbox permissions and any permissions on non-mailbox objects—such as distribution lists or a mail-enabled user—are not migrated. (i.e. A user mailbox migrated to Exchange Online that has Send-As, Receive-As, and Full Access directly applied to it will have that permission migrated. Inherited permissions will not be migrated. Also, an online user will not have access to edit a distribution list if that distribution list is homed on-premises, and vice-versa)
- Exchange hybrid deployments support the use of theFull Access mailbox permission between mailboxes located in an on-premises Exchange organization and mailboxes located in Office 365. A mailbox on an on-premises Exchange server can be granted the Full Access permission to an Office 365 mailbox, and vice versa. For example, an Office 365 mailbox can be granted the Full Access permission to an on-premises shared mailbox. Users need to open the mailbox using the Outlook desktop client; cross-premises mailbox permissions aren’t supported in Outlook on the web (formerly called OWA) (i.e. A user mailbox migrated to Exchange Online with Full Access can access an on-premises mailbox, and vice-versa)
Note: Users might receive additional credential prompts when they first access a mailbox that’s in the other organization and add it to their Outlook profile.
- We don’t, however, support the use of the Send-As, Receive-As, or Send on behalf of mailbox permissions in hybrid deployments between on-premises Exchange and Office 365 organizations. (i.e. A user mailbox migrated to Exchange Online with Send-As/Receive-As cannot access an on-premises mailbox, and vice-versa )
- We also don’t support delegating permissions to a mailbox or individual folders using the Outlook client. These permissions are only available when both the mailbox granting the permissions, and the mailbox receiving the permissions, are in the same organization. Any mailboxes that receive these permissions from another mailbox need to be moved at the same time as that mailbox. If a mailbox receives permissions from multiple mailboxes, that mailbox, and all of the mailboxes granting permissions to it, need to be moved at the same time. In addition to these permissions, the Auto Mapping feature is also unsupported when used between mailboxes in the on-premises Exchange and Office 365 organizations. (i.e. A user mailbox migrated to Exchange Online cannot be assigned permissions via Outlook to an on-premises mailbox)
Formerly, cross-premises permissions didn’t work at all. This changed in March 2016. See https://blogs.technet.microsoft.com/mconeill/2016/03/20/shared-mailboxes-in-exchange-hybrid-now-work-cross-premises/ . The situation improved 6 months ago but not everyone keeps current with this level of detail.