I understand that data transferred between servers over the internal network is not chargable.

What do I need to do to ensure that communication between two servers is happening in this way? Presumably I must connect to Server B using its Private/LAN Network IP (after adding one in the control panel) - but is this all that is required? (I don't know if I must do anything on server A to make sure that requests to server B in this way are routed over the Private/LAN Network).

Thank you!

If you connect to server B using the private IP, then that's the route that will be chosen. Also remember that after requesting the private IP in the control panel, you'll also have to add it to the server's own network configuration (configuring your server's network settings manually, see Linux Static IP Configuration in the Linode Library), and reboot that server for the changes to take effect.

Once the server has been rebooted with the new network configuration, you're all set!

Thanks NeonNero! I'll read that document.