I ride around here a lot and have a friend that works for the AMC. I'm not positive but I think he said its not illegal to ride the trails. He was a caretaker at lake of the clouds huts on Washington. I'll ask him. There are some good rides around there too. The RedTail trail in North Conway and also the quarry there is fun. Sticks and Stones is fun too. The last three places are all connected so you can do all of them with out going elsewhere.
If you go to a bike shop or someplace like ems they sell mtn bike specific maps for trails in the white mountains. I have the maps for the Rob Brook Area and the Conway trails. Those places are about 30 min south of Mt Washington. The ski areas like Attitash have down hill stuff that you can take the lift. I think they require a full helmet with face gaurd though.
If you like really technical rocky trails i highly recommend hitting Ft Rock in Exeter NH on you way there or back. The trails are on par or better than Blue Mountain.