Oddly, I had the "sale" link alone (no private sale), then futzed with the private sale URL I saw in the original screen shot to no avail. Eventually I saw the facebook link, clicked it, and forever after had the private sale as well. I didn't have an account previously, and private browsing or not, the private sale never showed up until I used the FB link. So I didn't have access, and then I did... I don't know the how or why behind the scenes.
That was sort of the case for me as well (I didn't have access, tried a bunch of stuff, and then all of a sudden did have access - but I don't know what it was, specifically, that caused the private sale to appear)