Your Echo knows how to connect to your router. It teaches the nearby Echos how to
That's not how it works, that would be an obvious risk if they gave the connection info to a different device and that device is now fully on your network, and they also wouldn't be able to control that it shouldn't use a lot of Bandwidth.
Instead, they use connection from Echo device to Echo device (using Bluetooth single as well as different unused wireless spectrum) and on device acts as a bridge to send the other device's info/alert/etc. Sort of the way Apples Find My work by utilizing Apple device to Apple device connection.