Several suggestions in this thread are acceptable. As long as the Bitwarden organization does not have the capacity to unlock your password database in any shape or form (other than by pushing out a rogue client), I am OK with this feature. To do this, the keys must remain with the user and the emergency recipients.
Note that granting access to non-Bitwarden users who possess no secret keys would NOT be possible with a secure approach if the server manages the recovery process. At a bare minimum, the recipient needs a secret key on a piece of paper. Other approaches, such as a private key stored in the recipient’s Bitwarden account, can also be implemented securely.
(After some more thought, it would be possible to implement this in a less secure manner as long as it is completely opt-in. If this is the case, whatever recovery key is used should be generated on the client side, not the server side, and it should not be derivable on the server side. Clients that have not opted in to this feature would simply not generate the recovery key, and of course these opt-out clients would never send any private key to the Bitwarden servers.)
Edited to remove incorrect information about LastPass.