“Headset and Speakers” function not working with game chat.

You'll have to enable chat output to be both headset and speakers. To do so, go to Settings -> Display and Sound -> Volume. In the Party Chat Output select the option of Headset and Speakers.

Azrael, the solution you provide is a lackluster solution. I stated the reason why above but so you don't have to go all the way to the top and read it again, I will re explain the issue.

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The xbox one does not have the same feature that the xbox 360 has, which is the chat through speaker and headset. So, I have a polk audio Melee headset but it is connected to my xbox one with an Optical audio and a USB cable. this powers the headset and
gives it audio from the game and chat. The xbox one looks at the optical out as speakers as most will use this port for running a sound system or something along those lines.

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Now when you connect the xbox provided headset, it takes the chat from your team mates that used to go through the speakers and runs it through the headset, being the xbox one provided headset. What this means is the headset I am wearing to hear the game,
the Polk Audio Melee that the xbox one registers as speakers, no longer gets game chat.

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If xbox one had the play through speakers and headset option, there would be no issue, but they don't so I can either put the xbox one headset on and not hear the game, or put the one headset around my neck, and let my teammates hear me but me not hear them.

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So your solution did not actually work.

I don't think it is possible unless you use a Kinect. But some games such as Destiny has this option in their setting to bypass audio from headset to TV speakers, however this only works for party chat & not game chat.

Got chat to stop going through speakers. Chat now works with the provided headset:

Be sure you are logged in at the top of the screen by clicking your player card icon.

the problem is if you plug in the included headset to your controller, it''s going automatically relegate all chat to that channel, sending and receiving ...when you don't have a mic plugged in, it relegates chat receiving to the same audio channel as game
sound, i.e. you hear chat on your TV, speakers, or headset but the kinnect is the mic...but a as you plug something into the cotroller, then both sending and receiving chat is relegated there, so you no longer hear chat on your TV, speakers or premium headset