I mean you have to follow the account and be accepted to see anything if it's private (some only have it set that way to avoid spambots and such, and will accept follows from any legitimate account). Beyond that it's the same as downloading Stories from any other account as long as you're using something that uses your logon (so, a browser extension or something like 4K Stogram that lets you sign in)
how to download private 24 hour stories?