The circle device must be within 10 feet of the router so it's impossible to connect to your neighbors router (assuming your not in an extremely tight apartment situation). You also can't walk into a coffeeshop or hotel lobby and use it.
Yes, you can monitor you children's internet activity but isn't that what you should be doing? As far as an open wifi network; you always assume that someone is monitoring your activity. Be security conscious and conduct sensitive business through a VPN.