I remember this being an issue a while back, but I don't think it is fixed as it still effects me. Details:
Whenever I disconnect from Destiny or lose internet connection to my Xbox, I lose the ability to receive invites or communicate in fireteam chat with non-Xbox players. In order to fix this issue, I need to go into the Xbox "Online Safety & Family", disable any one cross play setting, reenable it, and then I am able to connect and communicate with non-Xbox players in Destiny. No settings or switches get flipped off on the Xbox end, everything is always still set to allow cross play when this happens and I check the settings. This only happens in Destiny, I do not have this issue in any other game, and I remember a similar cross play issue with Xbox being investigated a while back. As far as I can tell, it has not been fixed, thus I am reporting.