I'm making my way through the Stephen King bibliography and about to take on "Desperation" which was specifically timed to be released the same day as "The Regulators" written by (at the time of publication) his already exposed, superfluous pen name: Richard Bachman. If it's anything like the very mild duality of Gerald's Game and Dolores Claiborne then it won't matter much either way, but I'm curious to know what those who have read both would recommend.
I took on Desperation first for no other reason than it was readily available at the library by my office. I blazed through The Regulators as soon as I finished to get the freshest perspective. I don't want to ruin either by giving spoilers but I think I chose wisely (or rather, got lucky). My personal reasoning goes for the fact that The Regulators had a few extra main characters that did not exist or had very bit parts in Desperation. I feel it would have been a distraction, expecting any one of them to appear before the last few chapters of Desperation.
TL;DR - I recommend reading Desperation first.
I don’t think any order will make a difference but I like Steve’s logic so I’d say Desperation is probably best. As for Valorum’s reasoning, I’d say it’s flawed. Yes, these books are read left to right, but they are also read front to back. Desperation was the front. The strong implication would be that you would read from front to back, e.g. starting with Desperation first.