The only method that I know is too inefficient: click on each profile and then see whether that profile is public or private.
There is no simple way to do this using the UI. You'll have to make use of the Instagram API to get the list of users that like a image and then lookup each account to see if it's private.
Another way is to create a tool that crawls the Instagram site, i.e. you give it the url to a post, it navigates to each user that liked it, and it checks if they are private.
There may be existing apps or Chrome extensions that do this.