As far as I know WP isn't supporting manual IP configuration, so there's no way looking for your so called 'local address'. If you're connected to a network via wifi your phone gets assigned an temporary IP (through DHCP).

To see the network-specific assigned IP address (under WP8), go to the "Wi-Fi" settings on your phone, hold down your finger on the appropriate network and select "Edit".

I didn't know if this question is still active.. There is two options to find the local address on your windows phone..

1. If you use a wireless connection, which the wireless router are set to the DHCP that you will get an automatic IP assigned by a wireless router. Here's the way to view your local IP :
   • Open your settings
   • Tap to Wifi
   • Tap and Hold your AP connected to then select edit

You will see the local IP your connected to

2. If you use broadband connection like 3.5G or 4G connection, you still can find the local IP address. To find the local IP address, you just need to download tools speedtest.net by ookla on your store. Here's the way to view you local IP :
   • Assume that you already have speedtest.net by ookla
   • Tap start, to test your current connection, when it finish
   • Tap settings, you will find your internal (local) IP and you public IP that assigned by your provider.

AIDA64 can show you the local IP of your Wi-Fi or mobile data connection, in addition to a host of other details.