Since the last patch, there is a random issue where the person joining the game will appear not to be part of the party. He will not receive party messages, and he will not be able to send either, receiving an error about not being part of the party.
If that player leaves the game, it will take a very long time before he actually leave. In the menu, he won’t be able to switch character. Should the host leave the game and restart a new one, he will automatically join it even though he wasn’t seeing the party.
Leaving the game is required to get rid of the party, but then the issue will probably crop back up right after joining your friend. This issue occured to me and some of my friends over the last few days.
Thank you for looking into it, and feel free to ask for any further details.
BTW, I tried restarting battle net too in case this issue was related.