The main cause of an echo on Xbox One party chat is due to the Xbox One being set to output party chat audio to both speakers and the headset at the same time.
To change this setting:
- Press the Xbox button to open the guide.
- Select Settings.
- Select All Settings.
- Select Display & sound.
- Select Volume.
- Select Party chat output
- Select Headset
You should no longer hear the echo.
If that doesn't work, please see more suggestions in this FAQ: