I'm one of these paranoid parents. I grew up completely free-frange out in the sticks and had several near-death experiences before moving away to a different town at age 6 & 1/2:

1) was hit twice by a car while riding a bicycle
2) got buried in snow while walking back home
3) almost drowned
4) while playing hide and seek, fell on a broken bottle and nearly bled to death

Given that my DD may have inherited my propensity for getting into trouble, I'm keeping her within my sight for a long, long time.