In order for a public user to edit a secure feature service, they would need to know the username/password to access the service. You optionally can register the service with ArcGIS Online by going to Add Item > On the web. When you specify the REST URL to a secure service, you will be prompted to store the credentials with the service, or have the users be prompted:
Whether the user is prompted for the username/password, or it's stored with the service, that is the user that will be recorded when a feature is created/updated, regardless of what ArcGIS Online user account they are using. The same holds true when using Collector.