Private tables can be used to troubleshoot or control intermediate results of transformations, as they are only written when needed. In order to do so, set the configuration mode to "verbose".
Despite the verbose mode being selected, no private tables are being written into the database.
Change the execution method of the task from "Agent execution" to "Server execution" and restart the task.
The agent has no direct access to the database due to security reasons, which makes it unable to access the data and write the private tables. The server, however, can write the private tables as expected.