When you say all audio on the host side disappears, I assume you have "Play audio on host" checked in the host's Remote Play options? (Steam -> Settings, Remote Play, ensure that advanced host options are shown, and check "Play audio on host.")

If you do and the audio's still going missing, that sounds like a bug and more information (is this Windows 10, Windows 11, some flavor of Linux -- and if so, which flavor of Linux, and what audio engine is being used) would probably be useful since that's something that should be fixed.

If the "Play audio on host" is checked and you're on Windows, as a workaround you should be able to go to the Steam Streaming Speakers when streaming is active and turn on "Monitor" mode, pointing it to the local device, which will cause Windows itself to mirror the audio there.

If you're on Linux, as a workaround the streaming speakers *should* have PIpewire monitor output nodes that can be connected to the other audio device in a Pipeway patchbay tool.

I'm having the same issues as OP streaming from Steam Deck to the Steam Link hardware. In my case the audio from Steam Link hardware plays fine but no sound plays through the Steam Deck (headphone jack plays no audio as well). When Steam Deck is set to play audio from its internal speakers, Steam Link audio is then heavily delayed and will cut out every few seconds. This wasn't a problem before until recently.