What permissions do I need to enable to allow public browsing to my on-Premises SharePoint Website?
I built it, MasterPage and all and now I need to be able to browse it from other computers...if even to start with other network computers.
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As I understand from your question you need to publish your SharePoint site to the internet that requires a cooperation with your system / network administrator and SharePoint site administrator, it's not only permission settings as you think.
So I tried to list the main steps that help you To make SharePoint accessible from the Internet,
claimsbased on your requirement
See also similar thread SharePoint 2013 on Internet