I can't find any other post besides mDMZ's original one which offers a working solution to this problem! The directions are for XB1, but someone on PS4 said they were able to do something similar.
If you can't play with family & friends on your home network, this procedure works every time so far for us. If you receive the "unable to join fireteam" error, the person who received the error should follow these steps:
Click the Xbox button to minimize Destiny. Click the Start/Menu button and choose Quit. Launch Destiny again. Often an error saying "unable to connect to Destiny servers" will appear, with an option to Dismiss the error. This will kick you back to the click-A-to-start-Destiny screen. Click A. You should now be able to select your character.
You will now be able to join a fireteam on your local network! You can even switch characters and rejoin each other. However, this process will have to be repeated each day. Somewhere between turning off the console for the night and starting it up again the next day causes Unable to Join Fireteam errors when you launch Destiny the next time.
LynaiyaNi21
EwoksDontBlink
WOLVES FUR EVER
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Because unless you have both systems port forwarded the Xbox changes the IP address when it restarts. The only way I've been able to successfully connect without port forwarding is to reset both consoles then resetting my router with the factory reset button. You will need a pen or something similar to be able to push this button. Be warned however that this wipes the router, including any saved settings and passwords.