Hi,
In order to better understand the situation, could you please provide more information,
- Do you have Exchange Hybrid setup with AAD sync
- What exactly it means when you say different forest - do you have multiple AD forests & domain
- How are the shared mailboxes migrated to Office365
Exchange 2010 is out of support. Please upgrade to Exchange 2016.
Cross forest permissions for Exchange hybrid supports Full Access and Send-As. If you move only the Shared mailbox to Office365 in exchange hybrid environment and the user mailbox still stays in On-premise, consider re-assigning the permissions and verify.
If the above suggestion helps, please click on Accept Answer and upvote it.