We have appexchange that contains thousands of apps which have been created by users to distribute publically. To understand how you can have an app listed in appexchange, read the following: https://www.salesforce.com/appexchange/faq/
We're about to launch an app that tour guests use to 'check-in' for a tour. It works in Salesforce1 using one of our user-logins; but it would make more sense for it to be a public app. I've toyed with the idea of adding the pages of the app to a Salesforce site, and then the app would be used with the site user permissions, but haven't actually tried it out yet.
https://www.salesforce.com/appexchange/faq/
Just something you could look into.