Chat on headset and game sound on TV. How does it work?

I have recently got a PS4 and wanted to ask how to set up that the chat via a headset (only use a cheap telephone headset) and the game tone is played on the TV.

If I keep the PS button pressed, I find settings for sound / devices that only allow chat audio to be set via the headset, but unfortunately this does not work. If you select this, the headset no longer emits any sound.

Maybe one of you can help me.

It already works as you described. So you can choose whether only the chat audio or all the audio is output via the headset.

However, this only affects the party chat, the in-game chat counts as game audio.