Yes, the owner of a Wi-Fi network can see the private IP address assigned to your device the moment you connect. This information appears in the router's connected devices list, often alongside your device name and the amount of data it has used. This is a normal part of how local networks operate and is not something unique to any particular router or provider. It is worth knowing that a VPN does not hide this from the network owner, since a VPN encrypts and reroutes your internet traffic but does not stop the router itself from seeing that your device is connected and communicating locally. A VPN protects what you do online from your internet provider and outside observers, but the local network administrator can still see your device on their network regardless.