I was told by one of the guys at Burlington Scuba (I believe it was Butch) that someone had asked a diver to look for a ring (or something) that they had lost in the public beach area. They were able to do it, but it required a permit which I believe was granted because of the lost item.
I don't know that they allow diving regularly there. Also, the only area where you can get a car close to the water is at the boat ramp and they do require that you have a boat with you (no power boats in Walden). The main parking lot is accross the street and you will need to walk a few hundred yards and down a hill to get to the beach area, I believe the deep water is on the other side (I've heard it is somewhere in the neighborhood of 90', but an online topo map should be easy enough to find).
If you do find out that you can dive, let me know. I live in Bolton (moved out of Concord about 6 months ago) and I have an inflatable which qualifies for the boat launch.
Good luck!