Problem

You have a hybrid Exchange environment and you need to grant SendAs permissions for a user on a mailbox.
The location of the mailbox matters here as the commands differ depending on whether the mailbox is in the cloud or on-premises.

Note:
You must have the correct Exchange role permissions to apply mailbox permissions.

Solution

Important. You must have the Exchange role access that allows this.

Exchange Online
Add-RecipientPermission `
    -Identity "Fakey Shared Mailbox" `
    -Trustee "Fakey McFakerson" `
    -AccessRights SendAs 
Exchange On-Premises
Add-ADPermission `
    -Identity "Fakey Shared Mailbox" `
    -User "Fakey McFakerson" `
    -ExtendedRights Send-As