They no longer give out pagers. Instead they take cell phone numbers. I suppose if a parent didn't have a cell, they might still have some pagers to give out. That's a rare occurence anymore, I'm sure.
You aren't limited by distance but rather by time. You have to be back by midnight or your coach turns into a pumpkin. I wouldn't feel comfortable leaving Disney property with DD in a Kids' Club, but I would be fine going to another resort for dinner, to a park, or to DTD. DD went to the Cubs' Den twice on our last trip, once I went to MK for a couple hours, and the other time I went to Epcot.