There can be problems when you access a site from 2 domains ie one public and one local. I would not recommend doing that, I would make all users use the public address so that all requests are processed in the context of the same host name.
You may also have a problem like that if you are using some kind of proxy server and the web server is not seeing the correct host name in web requests. It can work with a proxy server as long as the proxy server is forwarding the correct host name header from the original request.