The Gemara describes two Amoraim whose faces shined to the point that others noticed it. What is this glowing quality that certain saintly people seem to have?

It is a subjective sense that appears to be universally experienced by all humans who encounter the rare, saintly individual. Artwork through the centuries depict saintly individuals with some kind of glowing halo of light around their heads. We might argue that the Christian depictions of saints borrowed from Jewish conceptions such as our Gemara or Moshe’s glowing countenance (Shemos 34:29), however the ancient Japanese, Chinese and Egyptians also depicted various forms of Halos on Godlike or saintly entities https://en.wikipedia.org/wiki/Halo_(religious_iconography) 

The occurrence of this representation and its spiritual connotation throughout many cultures and eras validates its objective empirical reality. That is to say, collectively speaking, humans are picking up on something that seems to emanate from the right kind of person. Is it a tranquil and open looking face? Is it wise eyes (whatever that is supposed to mean?) Years ago I met a frum co-worker who happened to have been a medical professional working in a hospital where the prolific author, intellectual and science fiction writer, Isaac Asimov was spending his last moments on Earth. He told me that when you saw his eyes, you just knew you were looking into a person with an unusual soul. Was that a projection on my colleague’s part or was he seeing something that just cannot be fully put in words?

Holiness is something we see in a person's face, and you know it when we see it.  It is real, but how to get it, well that is somewhat of a mystery.

Translations Courtesy of Sefaria, except when, sometimes, I disagree with the translation

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