Error 'Insufficient Privileges' when email is sent from a Case

Resolving Insufficient Privileges Error When Sending Email from a Salesforce Case

To resolve this error, ensure the Contact associated with the Case is linked to an Account, or verify that the user has at least Read access to the Contact record.
Option 1: Associate the Contact with an Account
Navigate to the Contact record and add it to an Account. Once the Contact is associated with an Account, it is no longer treated as a private record, and users with access to the Case will be able to send emails and log activities.
Option 2: Grant the User Access to the Contact Record
Ensure the user has at least Read permission on the Contact object via their Profile or Permission Set.
Steps to Replicate this Scenario (for testing in Sandbox or Developer Org)

1. Log in as a System Administrator.
2. Set the Org Wide Default for Account to Public Read Only and Cases to Public Read/Write.
3. Create a new Contact and do not associate it with any Account.
4. Create a new Case and associate it with the newly created Contact.
5. Log in as a Support User and open the Case.
6. Click Send an Email in the Activity History related list.
7. The Insufficient Privileges error will appear.

See also:
Edit Object Permissions in Profiles 
Send Email Action Considerations for Cases