You should worry more about the toilet door handle, or the remote control in your hotel room.
I mean there isn’t too much of a health risk with headsets - you share all that dead skin sweat etc. at home with your family.
You can clean the aircraft controls and buttons with a sanitising wet wipe, (but who ever cleans the overhead panel controls or the cockpit door handle?), and can do the same with the company headset.
As far as speakers are concerned, with two audio sources, there is always the possibility of cancellation or amplifying effects, and if there is a time delay between the two sources, there could be echo effects ranging from a change of perceived ambiance, owing to a very short delay, or if digital processing is involved then there could be a longer delay which might affect the intelligibility. I personally have never suffered, but I do sometimes accidentally leave my speaker on, and as soon as the other pilot asks, I apologise and turn it off.