It used to work earlier but it might have been changed due to an Unity update at some point.
However we changed it now that it does not rely on these settings surviving the level.
So this will work again in next Patch.
Thanks for letting us know it was broken! :)

My issue is that I can not see how you were able to cause the missing needsatisfiers for one of the players.
I would like to know how to reproduce this problem but I fail.

I tried to place Lair in a similar way in a custom map and mirrored it, tested it then opened again changed a few tiles and mirrored it again.
But each time all worked as intended and the other player had correct needsatisfiers.
So I do not know how it broke for you.
Did you do anything unusual while placing the Lair tiles or while mirroring the map?

I could not try it my self as you removed the workshop you linked before (http://steamcommunity.com/sharedfiles/filedetails/?id=834570677)

How were these Lair tiles placed?

I did load the level and realized that the Lair tiles of the second player have no need satisfiers.
Needsatifiers are used to provide satisfaction to a need at a certain spot for a certain player, they are normally spawned when a room is placed.
However in the save game you linked above they are entirely missing for the Lairs of player 2.
However the Vault tiles have correct gold needsatisfiers for both players.

Something must have went wrong when placing the Lair tiles for player two.
Did you use mirror tools to copy the left side to the right?
If so an error must have happened which prevented correct parsing of the tiles.
Either that or the map got corrupted.

To fix the problem mirroring the map again should fix the issue.
And if not, can you upload your output_log after you mirrored the map, so we can see if an errors occurs.

Cheers

Next patch, changing owner of a titan in map editor should correctly remove it from old faction and add it to the new one.