To answer my own question, the salt ends up back where it came from, in the Gulf. There are a few technologies available for desalination, but the one used in the Gulf, where energy is cheap, is basically a distillation process: the seawater is heated to produce steam, which is then condensed to produce water. That's an oversimplification of a highly technical industrial process of course, but the principle holds good.
The problem (ask your local bootlegger) with any distillation process is one of diminishing returns in efficiency when you try to recover too much of the solvent (in this case, water). Above a certain concentration, the game's a bogey. This means that the byproduct isn't crystalline salt, but strongly saline seawater. Returning this to the sea is not without problems. The long term effect is a gradual increase in the salinity of the Gulf as a whole. The shorter term and more obvious effect is localised damage to the ecosystem through greatly increased salinity around where the brine is returned.
Armed with this knowledge, you might expect Gulf folk to be circumspect in their use of fresh water. But you'd be disappointed. The UAE is second only to the USA in per capita wastage.
Oh well, enough seriousness for one day.