PROBLEM:
Even if you have Allow Local LAN Access checked, your administrator can override the value and disable it on you.
WORKAROUND:
Assuming your local LAN is 192.168.1.0/24 (has IP addresses between 192.168.1.1 and 192.168.1.254), and you have admin rights on your machine, you can modify the routing table!
You need to do this EVERY TIME YOU CONNECT, as the Cisco client will inject the routes upon each connection.
This simply deletes the "override" by removing the route map between your local LAN range and the VPN Interface.
1. Connect to your Cisco VPN server
2. Go to Status > Statistics > Tunnel Details and Verify that Local LAN Access is "Disabled" under the Transport heading. If it shows Enabled, then you have another issue preventing your access which can't be solved here)
3. OPEN A COMMAND PROMPT AND TYPE "route delete 192.168.1.0" (without quotes, where 192.168.1.0 is your local LAN)
4. Try to ping or connect to a local machine to verify success.
Enjoy!