Local is relative, but if you're in Hudson/Pelham, there's Long Pond. It extends into Dracut MA, but has a public beach on the NH side.
I haven't lived over there in about a year, so I can't comment on the cleanliness/clarity of the water at this time.
The Nashua River has a few spots some friends go to bridge jump and what-not I guess, but I'm not much help there, as I've never been.
There's a few spots in Lincoln along 112, which, you just have to be on the lookout for...they're pretty easy to spot if you take your time to look. The two that come to mind are the spot under the bridge, on the right side of 112 heading north, right next to the hotel (I don't know the name, I'm going by sights...hopefully someone else can help), and Lady's Bathtub, which is between the condo complex on the right side, right before you hit Loon.
There are a couple spots in the Conway Area, most famously Diana's Baths.
There are waterfalls, and small pools the kids can swim in. There's usually a good amount of people there as it's a popular spot, so you might end up parking on the road, but it's not a big deal.
The Lower Falls on the Kanc are closed until November, but that's a good spot, as is the pool below the Covered Bridge.
You can swim at certain spots at most of the falls (ex.: swimming is allowed outside of 500ft from Sabbaday Falls) along the Kanc, so you'll have a nice sightseeing adventure, but you can't swim IN most of the falls.
If you're the hiking type, there's a five mile hike you can make to Thoreau Falls off the Zealand Falls trail, which has some neck deep (on a full grown adult) pools, and some small falls, atop one large fall. I would suggest a map for that hike, I've done it twice and it's quite fun, but quite long.