One (definitely not insurmountable) problem that would exist in such a federated and open system is credential authentication:
Currently, Apple signs your email address and phone number (hash) so that you can't impersonate somebody's trusted contacts and send unwanted material to them without their consent, which has been a problem for Apple in the past. That's supposedly also why they have removed the "allow all AirDrop senders" option in favor of one that times out after 10 minutes.
There would either have to be a federated alternative to that, or the open source system would have to drop sender authentication; then you could only receive AirDrops while your device is in "allow all senders" mode.