All personal setting for player 2 are not saved on our PS4 edition of Minecraft. Both players use a PS4 account. The inventory and last position are saved, but things like controller settings (invert look), Hints and skin are not saved. We also have a XBOX 360 and we don't experience the bug there.
Edit: After some further analysys, I think the issue is this:
Rather then settings not being saved for player 2, some of the settings are not stored with the players 'profile' or settings. When the players 1 and 2 switch, then the second player starts where the game was saved and has it's own inventory, but he gets the skin and controller settings from the player that was previously player 1.
So, it looks like some settings are stored with the players personal profile or settings, like location and inventory, and some other settings are stored with the "Player 1 Profile", like skin and controller settings.
Hope this helps narrow it down a bit.