Remember, to make your web app public, all components must be shared with the public:
Web Apps allow a user to share maps and related information in an interactive way rather than just a static display. You can take a shared web map and combine it with a Web App template to get a web mapping application.
See our blog post on Story Maps (from ArcGIS Online and other options), as well as a few examples that we've done:
See the next tab for a listing of Web Apps.
A link to a web app or a web app embedded in a web page or blog post will work for whichever group you've shared it with.
Journalism.co.uk has a post about Esri Story Map apps. The include some information regarding the creation of Story Maps using only a free ArcGIS Online Public Account (it is possible, with limitations):
You can create a Story Map for free with an Esri public account, but there are some limitations to what you can do with this type of access.
For instance, you need to upload your images and video to a third party website such as Flickr or YouTube to create a Tour.
ArcGIS only allows ArcGIS Online subscribers to upload images straight to the Tour, although public users can upload data from a CSV file.
To import images from Flickr, you need to first find the account you need, and then pull in images from a selected photo set or a certain tag.
A YouTube import works in a similar way, requiring map builders to know the username of the account in advance as there are no social media search functions within the import feature.
Note as well that Story Map will only import up to 99 photos or videos from each source when using a public account.
In a blog post regarding choice of Story Maps, Esri blogger Bern Szukalski also addresses what can be done with Story Maps with a free public account.