The Spotify Web Player does not currently have the Private Listening feature.
To use the Private Listening feature and all of the other more advanced features you will need to download and use the Spotify desktop app.
You can download the full Spotify desktop app here:
The desktop app also has a lot more settings listed on the settings page.
Help others find this answer and click "Accept as Solution".
If you appreciate my answer, maybe give me a Like.
Note: I'm not a Spotify employee.