Yeah, well, Mr. Smarty-Pants – before you coached me through the interview process so I could land my current job, I spent 14 years as a puzzle editor, so I actually know the answer to this from direct experience.
The job of a puzzle editor, put briefly, is to make sure that the solver has a good time. Some of this you’ve touched on already – checking that the puzzle solves correctly, for instance that it doesn’t break symmetry rules or include two-letter words [illegal in all but the cheesiest TV-Guide caliber puzzles]; that the “theme” entries are all correct and on-topic; and that the “filler” doesn’t include too much crap (obscure words, misspelled words). It might be necessary to rework the grid to get rid of some of said crap.
Making sure the solver has a good time also involves that anything that the puzzle is in good taste; you’ll never see the word “NAZI” in a puzzle, or see “AIDS” clued in reference to the disease. Speaking of cluing: Another element is making sure the puzzle is clued at the appropriate level of difficulty – a person who’s looking for a challenging puzzle will be just as pissed off by easy clues as someone who’s looking for an easy puzzle will be by challenging clues. Keep in mind that constructing a puzzle – putting together the answer grid – and writing the clues are two completely different skills, and most people who are good at one aren’t particularly good at the other. (Merl Reagle is the most famous exception to this rule; Mike Shenk, a lesser-known but equally talented puzzle guy, is another.) (Mike Shenk, BTW, is one of the people behind the “New Yorker Cartoon Book of Puzzles” that came out recently – which is wonderful.)
So the editor is responsible for both checking the puzzle, and also tweaking it to make it as entertaining as possible.
A big-name editor – like, say, oh, Will Shortz – also has other responsibilities, as Jonathan Chance suggests – like evaluating the literally hundreds of puzzles submitted to him, a lot of which aren’t very good. You also end up mentoring new constructors – critiquing their work and suggesting changes and corrections. He also has to deal with mail from solvers, etc.
Syndicated puzzles are, in fact, edited – I know a couple of people who do it – but not by big name people, thus the lack of byline.