This is one of Dickinson's wisdom poems. While it is commonly said that beauty is in the eye of the beholder, the poet applies the same idea to bird song. The Oriole sings the same song whether or not anyone is listening. Those who do hear, however, make their own determination whether or not the song is plain or lovely. The sense of beauty lies within.
I like the way she frames the alternatives in the first stanza. To hear the Oriole sing is either a "common thing" or "only a divine." Note the "only." Dickinson is being playfully ironic. Wouldn't we all rather have a divine experience rather than a common one? Well, maybe not, but certainly regarding bird song we'd rather hear the beautiful song of an oriole rather than the thin threads of chipping sparrow.
It is the ear that determines whether the song is common or divine, "Rune" or "none." The awareness comes from within. Something like this is meant in some of Biblical teachings: don't cast your pearls before swine; "He who has ears, let him hear." Song and melody can be explained in terms of sound waves and vibrations, nerve endings and brain process. The sense of beauty and mystery come from something deeper within the human experience.
It may well be that Dickinson has poetry in mind as well as bird song. Poems might not have an intrinsic meaning; outside human experience they might be completely meaningless. Critics of her day might look at the elements of poetry: meter, rhyme, etc.; but that is like analyzing bird song. It is a satisfying exercise for the common but misses the divine entirely.