A Private key is a secret associated with the user's Arcana wallet. The private key is generated using a distributed key generator (DKG) component of the Arcana Auth product. It cannot be accessed by anyone other than the user.
Authenticated users can export their private key via the Arcana wallet UI securely. After exporting the private key, the user continues to be part of the Arcana ecosystem. The private key access for authenticated users is not disabled after exporting the key and the key shares are not ejected from the system.
In future, you will see a more robust ADKG subsystem in the Arcana whereby the key shares are periodically refreshed. This will further mitigate the risk of a malicious actor accessing a few key shares and reconstructing the user's private key.