Hybrid Permissions is when one user or resource is in one location and the other resource/user is in the other. i.e. a shared mailbox is in Office 365 and a user is still on-prem in Exchange.
Most permissions/delegation will continue to work correctly but some will not. The easiest fix is usually to complete migration to Office 365 for affected users as fixing hybrid permissions can be difficult or sometimes not possible.
Permissions Type | Works in Hybrid | Notes |
Full Access | Yes | |
Send-On-Behalf | Yes | |
Send-As | Yes* | Requires manual fix by mail admins, easiest to migrate all users where possible. |
Calendar Delegation | No | Editor/Author access is not reliable in hybrid scenario. |