I think the private list data setting is more about how other people in your organization can see/access information in the Directory. If you’re the one who created the list, you should be able to see the information even if it’s set to private. Maybe your Admin made it so that lists are private by default?
I have a few private lists in my Directory, and I can still use the embedded data from those lists in my surveys. I don’t do anything special/different because it’s a private list - I just add the embedded data variable name to the survey flow, like normal. The information from those fields shows up in my survey results (I do see it when I download the survey results)
If you’re hesitant and want to test it out before contacting real respondents, you could make a test private list with just your own email address and some fake embedded data, send yourself a test survey, and download the results to ensure the info you want is showing up in the results.