With Dolby for headphones on no sound come from TV or headset

As the page that XBF Javier R linked to explains, the console can't know if you use the headset for chat only or for game sound as well, and will not mute the TV sound output.

If you don't get game audio from your headset, but chat sound is working, then go to the settings, and under "Display & Sound" enable game sound through your headphone.

Unless you have bought the Dolby Atmos for Headphones app (and not only the free trial app), you should use Stereo or Windows Sonic to get game sound from your headphones.

If neither game sound nor chat sound is coming from the headset, try the headset on a different controller, or try different headphones on your controller (e.g. earplugs that came with your smartphone or something like that).

Had this issue myself, all you need to do, is go to the sound settings and in the mic, set it to headphones only, do not use headset and tv for the chat.

Set it to HEADSET only. this will stop the issue.

If you dont want to do that, you will need to use standard stereo, not Windows Sonic or Dolby, as this is were the echo can come from.

Sounds coming from headphones and TV at the same time is normal for the console...the other day I left my stereo on (my TV is wired through it) and couldn't work out why the audio from my headphones sounded different!

As to the audio through a headset being lower quality, that would depend really....if the audio signal for the headset is coming directly from the Xbox (wireless headset or wired), then I'd argue it's better quality than that carried to the TV via the HDMI cable. For me, my wireless headset is plugged into my controller (using an adapter) and the sound quality is miles better than the TV is capable of (is even better than my very good stereo too)

This will depend on the Make/Model of your Headphones. Most Headsets (such as Turtle Beach) include a cord that you plug in to the TV and plug the sound composite/component cables into the back of them that streams the sound. These styles of headsets also
have a headphone jack (3.5 mm) that can be plugged directly into most TVs directly as long as the have a 3.5 mm port for headphones. This will give you the same surround sound quality as the other method. If the sound does not come through as surround sound,
check your TV settings. Some TVs have separate sound settings for using headphones, and it may need to be adjusted to allow Stereo Surround sound.

If you’re plugging stereo headsets into your controller or your console for the first time, your headset format defaults to
Stereo uncompressed. You can use spatial sound with any headset by using one of the following:

Windows Sonic
– This is the spatial sound setting Microsoft offers for an immersive headset experience. Plug any stereo headset into your controller or plug in an Xbox wireless accessory to your console before changing your headset setting to
Windows Sonic for Headphones.

Dolby Atmos for Headphones – Dolby Atmos for Headphones is a premium headset audio format that works with all headsets. If you’d rather use Dolby Atmos for Headphones, choose this setting
under Headset audio. You can learn more about Dolby Atmos for Headphones in the app.

To change headset audio settings:

1. Press the Xbox button
   to open the guide and select System > Settings.
   
2. Select Display & sound > Audio output.
3. Choose one of the headset audio options.