In this poem, Death is often described as the Narrator's escort. In any case, they are fellow travelers. In the grand scheme of the poem, it would be more accurate to say that Death is the Narrator's fellow traveler or even a guide but it does seem that Death comes to "call on her" as a potential suitor would in Dickinson's time. And Death does certainly seduce the Narrator, first appearing as a kind and civil gentleman. It isn't until the end of the poem that the Narrator realized this seducer was/is Death. In this scenario of Death posing as the Narrator's potential suitor, Immortality could be their chaperon.
One interpretation of why Death and the Narrator do not speak to each is that there is no need to speak. Death's seduction is ineffable. Words are unnecessary. As a spirit, perhaps Death's means of communication/seduction are revealed via his gestures, such as his gentle yet seductive manner of carrying the Narrator through visions of life. The significance of the scenes they pass and the pleasantness of the ride have seduced the narrator to a point which, like a sedative, she is comfortable and satisfied enough that she also feels no need to speak. It is clear in the second stanza that she is comfortable with Death and their pace:
We slowly drove - He knew no haste
And I had put away
My labor and my leisure too,
For His Civility -
Another interpretation as to why they don't speak is that the Narrator is dead and Death is metaphysical. Being beyond the physical, perhaps speech is communicated via psychic energies. It seems that they are also outside of time. Centuries felt like a day to the Narrator and she notes that the Sun passed them, indicating they are moving incredibly slowly. In this metaphysical, atemporal journey, any communication might be telepathic or intuited.
In the end, the general implication is that speaking is not necessary. Either Death is the Narrator's personified spiritual guide or Death is an actual spiritual guide from death to the grave. In either case, since there is no conversation in the poem, the indication is that speaking is not necessary. Death has done this before and he therefore knows how to seduce so well that he doesn't even need to speak. In other words, Death's skill at transitioning a soul from life to Eternity is so practiced that he can seduce a soul without saying a word.
This is fitting because the speaker/Narrator notes at the end that she "surmised" the "Horses' Heads / Were toward Eternity -". So, another reason Death does not speak is that the departed soul (Narrator in this case) is meant to come to the realization of her death on her own.