Yes and no. The showrunner will do a final pass for voice and tone ("The Doctor wouldn't say 'It's there,' he'd say 'There it is.'"), but it's not obligatory that there'll be any changes, or any substantial ones (I believe Russell T. Davies mentioned in his book that he didn't or hardly rewrote Moffatt's Doctor Who episodes, for instance). For earlier drafts, it's possible there will be notes sent back for the original writer to implement, though considering the hectic pace of television production, it's common for the showrunner to simply rewrite it themselves.
In this case, an example I remember is that in Gaiman's first draft of "The Doctor's Wife," the Corsair was much more explicitly described as a proto-Doctor, who was the Doctor's childhood hero who he modeled himself on, which Gaiman ran by Moffat before he'd even done his first draft (Moffatt thought having a "Gray Ghost" to the Doctor's Batman wasn't the right choice, and the Corsair became a peer instead of an idol).
There's more information about the writing process here on a Q&A Gaiman did on his blog, but since I doubt any of us are going to get the chance to do a Writer's Guild credit arbitration where one of us is sent every draft and correspondence between the two of them to figure out which word came from which person, the idea that the episode will be fine if it's more Moffat's than Gaiman's feels a bit like debating angels dancing on the heads of pins. Does it still count as Moffatt's creation if it was his feedback but Gaiman did the actual writing?