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Most people on this forum do games and video editing, don't bother with most software recommendations, it just complicates everything and you'll NEVER get rid of the latency, here's equipment that can help
 
https://focusrite.com/en/usb-interfaces
 
https://www.presonus.com/products/Studio-1810c
 
https://mackie.com/en/products/mixers/vlz4-series/402VLZ4.html   OR  https://mackie.com/en/products/mixers/vlz4-series/802VLZ4.html
 
( If you need more complicated monitoring ) https://www.artproaudio.com/product/headamp-4-pro-five-channel-headphone-amplifier-with-talkback/  
 
And at least TWO of these BALANCED cables for long distance, I'm not sure how far you have to go, these are 20ft(might even want to look into cable snakes which is multiple back and forth cable options)
https://www.guitarcenter.com/Mogami/Pure-Patch-1-4-1-4-TRS-Cable-20-ft-1369238774453.gc?pfm=krecs+1369238774453+30144+GCWPDPSI
 
 
 

Ok a little more serious. in todays world.
I live in China and we use wechat which i believe funcntion wise is exactly the same as Whatspp. assumption on my part as I reside in Southern China.
Anyways. my wife is often using her mobile and using wifi connection is connected to her friends. bluetooth on the phone sending the audio to a decent quality speaker and whilst she or they play games watch movies etc they just talk to each other....they sit and discuss the movie they are watching and often assist each other in games etc...and these others are not down the hall but in other cities....so i suppose rock on technology as an intercom is now definately prehistoric.
If this issue you have is merely communication with utmost ease and no lag with your partner this already get you 90% of the way. the last 10% is either bluetooh headphones instead of speakers or a drop in item that captures the bluetooth signal but has an output for your headphones. Oh and my wife is often talking laughing with not one but sometime 5 or 6 people so I have a gaggle of geese in my room at times.

Do you have any buffering on your TeamSpeak clients and servers set to their absolute minimums?
 
I can think of a few ways you could do this, using tools like Dante Virtual Soundcard. What's your budget and how much of the hardware in your diagram do you already have?

years ago when hardware was no where as efficient as it is in todays world my son and i made our own solution for back in the days when playing Duke Nukem 3D Atomic addition was a thing. yup years before VOIP etc etc etc.
we simply used an old office intercom solution where I ripped out the speakers and wired in headphone jacks. also had someone assist so that both ends ecieved as well as output but at the same time. the salvaged office intercom was push to speak or passive sit and listen.\Anywas it worked and he sat upstairs in hs room whilst i was downstairs and we networked Nuke Nuken 3D and had 1 on 1 fights...........awesomeness to the extreme..well for 1996 anyways but i have a feeling this is not the solution you were looking for ?
A little better than yelling down the hallway though.

I made it on Diagrams.net
here's a link to a version of the diagram you can edit - just click the pencil icon on the bottom and it'll open in the editor.
Thanks for taking the time!

Out of curiosity, what's the point of all of this ? 🙂
Can't you just rather shout across the office ? 😄
 

I'd try going about it in software with Voicemeeter Potato for a start, might be good enough already.

i feel like i exceeded my brains memory looking at this