Find the chat output
On Xbox Series X|S, open General → Volume & audio output and set Party chat output to Headset & Speakers. That sends party chat toward HDMI as well as the headset. This setting covers party chat only. Elgato notes that the Xbox sends game chat and lobby chat only to the connected headset, so to capture those you need a mix-amp or similar headset device with its own audio output, or a wired headset with a chat-capture cable such as Elgato Chat Link.
On another console, inspect its documented headset and HDMI behaviour. Do not transfer the Xbox setting to a PS5. Console party-chat routing explains the separate paths.
Check the receiving audio source
Ask someone in the party to speak while watching the capture source in OBS. If game sound moves the meter but their voice does not, repair the console or headset route first. If their voice moves the meter but the recording is silent, inspect mute and recording-track assignments.
Your own commentary microphone needs its own intended path to OBS. Do not assume that a headset microphone is included in the audio coming back from the console.
Record both speakers
Make a recording in which you and a party member speak separately. Listen for each voice, then test the audience preview. Keep the working chat path in your stream checklist.