^^^ Many (indeed, almost certainly most) crosswords are syndicated and can therefore appear in dozens of different publications. So questioners who refer to a local publication might well be being perfectly truthful, even though the same crossword also appears in a national one.
Compiling crosswords is a worldwide business. For example, I've seen questions asked on AB about crosswords appearing in Irish journals, where a quick web search shows that the same crosswords appeared in the New York Times, or similar, over a decade earlier.