I always sigh when some one says "I heard" and doesn't say who they heard it from, and what circumstances the all-wise adviser was in when making this judgment .... when he/she judged Naxos over-touristed what time of year did he visit? July? August? You do not say--and perhaps you do not know, you just take his/her word for it. Yes, more people are coming --- NOT because they "hear" there are luxe hotels, platoons of staff, swim-up pools, hot-rock massages, "greek nights" with men in white skirts -- no, because they've read that you can get away into wonderful hillside villages, walk on mountain roads, buy fresh local apricots & tomatoes at little markets... and a few miles inland, they can be on actual farms.
As for local communities & populations working at things other than tourism --- Naxos is the largest Cycladic island, with more farms than any other... biggest source of best Potatoes in greece... many vineyards (and no fawncy sampling tours). .. plus, the marble industry in Paros is kaput, whereas in Naxos, it's going strong, with 18-wheelers full of Marble rolling onto ferries every week . Overly touristic Naxos Villages? In Paros, the town of Naoussa has become something of a tourism village "picturesque", not any fishermen there any more, tho lots of brightly-painted boats tied up at its waterfront ... Both islands have "beach strips" with hotels for vacationers, but Naxos doesn't have a prettified village. Down to earth prices in Paros, vs Naxos?? I dunno... haven't been for 3 years, but in Naxos town still could get a hearty dinner for €11-13, more than you can eat, & a liter of house wine for about €4-5. Paros prices more reasonable? Then how can I get a balcony room with sea view in St. George Naxos @ €40 a night in June?
Sorry for belaboring this point, but I do think that you should trust people who do know each island. I have been going to Naxos since 1999, and have stayed there 13 times ... and it has not been "spoiled". Yet. Go while you can. Same for Antiparos.