Public networking in Vagrant can be quite a confusing concept. In essence, it is private networking, but Vagrant will attempt to allow public access from outside your host machine (if your provider and machine will allow it) instead of just allowing access from inside the host machine.
By performing the following steps, you should be able to access your Vagrant machine via an IP address from another device on your local network. Make sure that you have nginx installed so you know when you have successfully connected via HTTP to the IP address. I have been able to view the nginx default page using my smartphone on the same local network. If you were to use the private_networking option, this would not work and my smartphone would not be able to load a page or find the device, which would result in a timeout.
There are two main ways to set up public networking: you can use DHCP or manually assign a static IP address.