Hello and welcome to the forum!
I guess I could be called a "regular" to the Blackstone area. I probably camp/fish there every other year on average.
The campsites in the actual provincial rec area are treated like any other maintained campground. You have to pay for these sites (self-registration) but they give you the benefit of a gravel pad, nearby outhouse and table. If you are like most of the campers out there you will take the "user maintained" areas. These are first-come first-serve but you want to use common sense courtesy. For example, if there are many spots open you probably shouldn't set-up right next to somebody else. Just make sure that you leave the site as clean or cleaner than you found it and take a drive to find firewood rather than taking down trees or deadfall near the camping area.
The river is relatively safe for swimming/tubing depending on the ages of any kids. There are deep areas but they don't last very long before you are in a couple feet of water. As we know, the nice swimming holes are also the nice fishing holes. It is more-or-less accepted my fishermen that the holes close to the camping areas are going to see some swimmers. Usually, if I come to a hole where people are swimming I will just pass by to the next spot. If someone is already fishing there when you arrive, don't go swimming in their fishing spot
I hope this is helpful. It is a beautiful river - have fun.