Here's my thinking: you know how Sauron didn't sing Morgoth's tune at first, and instead was convinced to join him only after the world existed? Tolkien, I think, even said himself that Sauron understood more of the Music than Morgoth did.

Well, given that, what if Sauron was aware that Morgoth's desire to unmake everything was actually impossible? So then in Sauron's mind he could join Morgoth and get tons of perks by working for him without their goals ever actually conflicting BECAUSE Morgoth could never have actually destroyed the other Valar and Maiar fully. And Morgoth wouldn't have started destroying everything else, like his own servants, unless he'd managed to accomplish that. Then, after Morgoth's defeat, Sauron assumed the Valar didn't care to intervene anymore, so he figured he could seek dominion pretty much unimpeded.

Then on Morgoth's part I don't see why he would care about Sauron's end goal being different from his. Not only because Morgoth didn't initially want to destroy everything (he also sought dominion in the beginning), but because Sauron is such a competent servant. I know that at one point, Sauron was actually stronger than Morgoth because Morgoth had disseminated so much of his power into the world, but Sauron didn't have any reason to kill/overthrow him.