The Ma'or VaShemesh Maor VaShemesh, Parashat Balak explains that what Bilam observed was not merely a matter of physical privacy but of spiritual individuality: the principle that "every righteous person must hold to his own path and his own manner of holiness, and open for himself his own opening" — meaning that each person forged his own independent approach to kedushah rather than simply imitating the outward practice of his teacher or his fellow.
According to Maor VaShemesh, Parashat Balak, the fact that the entrances did not face one another signified that no one looked to walk through the opening that his neighbor had opened in holiness; each individual went through the entrance he had opened for himself — and it was precisely this that caused the Shechinah to rest upon them.
This reading finds a resonant parallel in Mishneh Torah, Human Dispositions 5:7, which similarly directs a person not to look upon his fellow at a moment of his failing — to avert his eyes — reflecting the same ethic of not scrutinizing how another person is living his life.