If each audio file is contained in a separate record, you can manage this by record-level access privileges. Have a separate user account for each user, and set up a privilege set that references the logged-in user name to decide whether to give that user access to the particular record. You would also need to have a field in each record that contains the user name or identifier of the user(s) allowed to hear that file.
Feel free to ask for clarification.
This isn't the issue, I'm afraid. The users only have an iOS device and they won't be running FM Go. I would like the desktop Mac to broadcast the stored audio file over Wi-Fi, so that each user can hear the audio file over headphones. They won't themselves have any access to FileMaker running on the desktop Mac.
I don't even think what you describe is possible. If the audio files are stored in the FileMaker database, I don't think you can make them accessible to any other computer or device without FileMaker Pro/Go.
As I read your original post again, it seems like you want to go to the record in FileMaker on the Mac, click Play on the container field and have the correct iOS device capture the sound stream. Sounds like dreamland to me.