We don't use either one of them. They sit bare in the washroom until some kind guest puts the roll on for us.
The reason we leave our paper rolls on the counter is that one too many times I have turned my head for a second only to discover a pile of paper knee deep and one (or two) very pleased girl(s).
I have found a ribbon trail through the house before. Despite what she said, it was not a treasure map.
The little one bats at the rolls like a kitten, delighted that it spins and come apart. She was also very delighted to put it all in the toilet and watch it shrink as it absorbed all the water.
So I say we can't be the only ones. We can't be the only family to leave their toilet paper rolls on the counter, out of reach of curious, devious little hands.
The counter method does pose new problems though. If you don't put it out of reach it can get knocked into the toilet, the sink or down the hall. If you leave it on the rim of the bathtub you are just begging someone to wonder whether it will sink or float.
I don't mind the new problems. I really mind the unrolling. So I guess that even if we are the only ones who do this, it works for me!