How to access my WP site locally and publicly? [closed]

Try accessing it using a direct IP address: Example: http://192.168.1.200/your_site_name

You may not even need the site name if it is the only site hosted from that server.

Ok in that case then you need to get the direct link from outside, replace the server URL with the IP address and bingo you should be there internally. The reason the link redirects you is because your WordPress settings use the for its page direction in the permalinks and general areas.