When the emails are being sent on behalf of the other user as a Delegated mailbox, the emails are placed into the Sent Items folder of a delegated user. The sent emails are not presented in the owner's (delegator) mailbox.
User delegation configured in Webmail settings -> Delegation section.
Once the delegation is configured, the necessary user folder can be viewed under the Shared Folders section. Once the original recipient receives an email, the delegated user is able to see the emails there.
Then the delegated user (vlad) can reply to the emails. Afterward, the sent email will be placed into the Sent Items folder with the addition of "on behalf of Full name".
The emails won't automatically be synced to the original recipient's mailbox. This is the expected behavior of the Kerio Connect and is considered as Functions as Designed.
The delegated user can forward the necessary sent emails to the owner or use CC or BCC feature when replying on behalf of the user.