Your company public profile, just like your user public profile, is the contact page that will be visible to all of your direct partners (from their own Contacts page).
You can access it and see it as they'd see it from the company settings:
You can also access it from your Contacts, by opening your own page. The home page of your own company contact is actually the same as your public profile in your company settings.
Note that only the first tab is completely public, everything else, most notably the scheduled activities and validation workflows have their own rules for visibility.
Except your company name, there are no required fields - if you do not wish to share contact/location information with your partners, do not fill your public profile.