It's not "impossible". But it's certainly not easy!
Up to my CPL training was all done
sans headsets - including controlled airpace flights.
My first instructing job was also without headsets. You do things such as configure the a/c for a lower power/lower noise before talking, brief adequately before flying (instead of using the flight time to brief), use single words or VERY short phrases instead of long(ish) sentences to pass information or instructions etc etc...
cheers
T
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WHAT?? SAY AGAIN!!