The concentrated minerals and metals in the waste, as well as the chemical additives used during the desalination process, are incredibly difficult to contain given their solubility in water.
There is a lot of desert where desalination is typically used. Seems like putting problematic water soluble in places where the main problem is lack of water would work out ok. Unless they are also blown around by wind in some problematic way.
Is this more of an issue for evaporated brine than for regular landfills? I thought that modern landfills had pretty advanced systems for preventing groundwater contamination, but it does seem possible that the problem is harder for water soluble brine.