The best solution I could think of was if a password manager like KeePass would support managed multi-user credentials. That is, each individual has their own KeePass keychain with their own personal passswords, but an administrative user can insert a special hook for a shared password. So the user could use their KeePass passphrase to login to the shared Twitter account, but they wouldn't actually know the Twitter password and it wouldn't be stored on their keychain. Any time they needed to login, their KeePass would authenticate itself with the admin KeePass, which would log them into Twitter for them. When the person quits or is fired, the admin can just revoke that person's access to the admin KeePass keychain. No need to change the password and email the new password to everyone (thus creating a potential security breach) because the person who left is a potential security breach.
LastPass supports this on their "Premium" and "Enterprise" accounts.
You can add sites to a folder which the administrator can control and that administrator can decide if the user will be able to 'see' the password or leave it hidden to all users.
Users will need their own unique password (and potentially Two Factor auth) to access the 'hidden' Twitter password account.