Can't hear other programs' sound when using Bluetooth headsets with voice chat applications on Windows 10

This annoying behaviour is caused by limited Bluetooth bandwitch when using both mic+headset channels. It's quite well explained here: https://www.howtogeek.com/354321/why-bluetooth-headsets-are-terrible-on-windows-pcs/

Here's the salient part for your issue:

Technically, when you’re just using the headphones as a sound output device, they’re using the A2DP Bluetooth profile, and ideally using AptX for maximum sound quality.

When you need the microphone, they’ll use the headset profile or hands-free profile (HSP or HFP). This allows for both recording via the microphone and playback via the headphones, but the headphone sound quality is terrible when using HSP or HFP.

If you’re just using a Bluetooth headset to take a phone call, that might be fine. If you want to speak into your headset’s mic while listening to music, playing a game, or watching a video on your PC, you’ll be frustrated.

For a PC headset with an integrated microphone, Bluetooth is a terrible choice. You’re better off getting a wired headset, or even just getting a separate microphone.

Bluetooth 5.0, which has already been announced, offers much higher bandwidth. This should let future Bluetooth headsets play back audio at high quality while the microphone is being used, in theory.

You need a new headset.

If you open up the sound settings you'll probably see something like this (image taken from the link in OP)

Notice the bottom check mark is Stereo, and the phone icon is Hands-Free. For microphones like this, Hand-Free also uses Mono playback, which typically sounds bad. It can only use Mono/Voice/Hands-Free or Stereo (no voice) at one time. If an application can use voice, it will typically turn off stereo but Mono won't work, and the sound will cut out. You need to set Hands-Free/Mono as default, and deal with the poor sound quality.

I am not sure if this is a problem with all bluetooth headphones or only ones with built-in microphones. I have bought 2 different headsets off amazon and they were both setup this way.

Here is how I get by: Set the default communication device as the hardware microphone and the output device as Bluetooth stereo for all applications manually. If your using windows 10 this is easily done in the settings application under advanced sound. I don't know the equivalent for previous windows OS.

I have the Bose Quietcomfort and solved the problem by using the cord that connects the headphones to the computer. When using Bluetooth I could only manage to chose between chat or game sound.

I had the same problem. I fixed the issue by setting the Headset as the default device (not the headphones). Sound quality sucks but it works.