The REST api must be enabled on your site, you then choose which "roles" have access to use the rest API.. The API itself leverages a users permissions, as such a user can do anything via the api that they could do on your site.
If you had a non user specific permission set in mind... IE like the same stuff that the anonymous user could read on your site, you could create a specific user to use for rest, give only that user an APIKey which would be embedded in your application, and deny that user write permission to your site... Doing this would have all the "Rest Widgets" logging in as this specific user, which did not have post permissions.
Dan