Sam as Mystic
I was once part of a group effort to "test" Frodo and Sam according to the Myers-Briggs Personality Inventory, so it's one classification system where I feel I'm not relying solely on my own judgment. In the two sets of Myers-Briggs characteristics most related to our discussion here, Frodo is an introvert (I)/intuitive (N), while Sam is an extravert (E)/sensate (S).
The majority of people, at least in Western societies, are ES's, like Sam. But there are some fields of endeavor where you run into a flood of more Frodo-like IN's, and one of those is spirituality/religion. Because of that, much of our literature on prayer, spirituality, and mysticism has been written by IN's, and comes from an IN point of view. I think it's safe to say that most of us would find it easier to see an introvert/intuitive as a contemplative or mystic than we would an extravert/sensate. But that's a stereotype rather than reality. The previous essay on "Everyday Mystics" notes that according to Catholic teaching every human being has what it takes to be a mystic. Ignatius of Loyola, who pretty much invented spiritual direction and had a wealth of wisdom about discernment of spirits, was probably an ESTJ, and Francis of Assisi, who was no stranger to spiritual ecstasy, seems to have been an ESFP. It may be fitting that when we look specifically at Frodo's spiritual development in the next section, we'll do so through the teachings of John of the Cross who was probably an INTJ, which would mean his personality was somewhat similar to Frodo's.
We get to know both Frodo and Sam from the inside out in LotR, and they're opposites in many ways, so Sam is a good character to use to look at another side of mysticism before we delve into Frodo's.
An extravert is more likely to process thoughts externally, by "talking things out," rather than internally as an introvert would, and a sensate relies on his own senses and on a step-by-step process instead of moving in "jumps" and assimilating information without quite noticing where he's getting it from, as an intuitive tends to. An intuitive might well use his senses to take in another person's body language, but decide "He's angry with me" without taking any notice that he's picking up evidence through his senses or how he's integrating it; his conclusion is something that he somehow "just knows." A sensate might be more likely to think, "Oh, oh - she's pursing her lips like she did the last time I forgot to clean the dirt off my feet before stepping onto the rug. She must be mad at me again," while glancing down at his feet to see if they're dirty and maybe even asking the question out loud. If there's no one else to talk to while he's going through a thought process, Sam will talk to himself. And he experiences his overtly mystical moments through his senses - at least, that's how Sam would describe it.
Frodo has dreams - although sometimes he's not quite sure if they really are dreams or an indefinable something else. On the other hand, during the night at Bombadil's while the other three hobbits are dreaming, Sam sleeps contentedly, "if logs are contented." Sam learns through his senses, so a dream might very well have less significance to him than it would to the more intuitive Frodo. For Sam, something that's "just a dream" might be brushed off much more easily than something he experiences as real.*
In a way, Sam's sensate nature makes his mystical experiences that much more potent. Frodo might experience something "either in his dreams or out of them," which puts a guard on how it should be interpreted, and in the material we have from the Red Book, he very seldom ventures an explanation. Sam, on the other hand, "simply" sees what he sees, with perhaps the best example being his thoughts as he watches Frodo sleeping in "Of Herbs and Stewed Rabbit." He ties the vision (what else could you call it?) to a similar one he had "...in the house of Elrond..." which also came to him through his sense of sight. No dream state is needed. Neither is any difficult or uncertain interpretation of what it might mean: "He's like that, and sometimes it shines through somehow." Does Sam realize that he has the eyes to see something that most don't? We're never told. And we don't know whether it was Frodo as he wrote this part of the Red Book, or the narrator as he rewrote the story for us, who joined Sam's experience here with Gandalf's thoughts back in Rivendell. Here as in other places in the book, it's not surprising that most of Sam's mystical experiences are centered around Frodo. "But I love him, whether or no," is the very state needed to see more deeply than the senses would normally allow.
But even though Frodo remains Sam's center, there's a change in the way Sam experiences this as time goes on. His deepest moment of realization comes when he sees what is most likely Eärendil's light in the sky after he and Frodo have already entered Mordor, as discussed in The White Star. I don't think it's an accident that Sam has this revelation after his brief time as Ring-bearer, an event that seems to play an important part in the deepening and widening of his spirituality.
While he carries the Ring, he's able to recognize and defeat Its temptation to want more power over creation than he legitimately has, which is the temptation the Ring gives to everyone, in one way or another. But it's perhaps more important that he actually experiences this temptation, which gives him a clearer understanding of others who face it as well as an awareness of his own limitations (he doesn't seem to fit Tolkien's description of him as "cocksure" so much afterward). And, although he probably wouldn't admit it even to himself at that point, he has more compassion toward Gollum on the slopes of Mount Doom than he did on the stairs of Cirith Ungol (although you wouldn't know this from the movie).
After the ship carrying the Keepers of the Rings leaves the Havens, Sam stands for a long time, letting the sound of the Sea become part of him. It's a sound that "troubled" Frodo's dreams long before the Quest, an unsettling awareness of the depths of the unknown that was part of what "graced and gifted" him for the task appointed to him. Sam's experience of the Sea seals his own fate, in somewhat the way it does for Legolas. Once received by someone who has been prepared to hear it (note that it didn't affect Merry and Pippin in the same way), it can't be forever escaped, any more than the Hound of Heaven** can be finally outrun. From the beginning, Sam had "eyes to see," and at the end of the story he has gained ears to hear, in part because of his time of being a Ring-bearer. So it makes sense for Frodo to tell Sam that it's because he, too, was a Ring-bearer that his time may come to sail West.
Besides his personality type, there's another difference that Sam brings into the mystical equation we find in LotR. As far as we know, Sam doesn't have a drop of Fallohide blood in him. Bilbo, Frodo, Merry, Pippin - even Fatty Bolger - have a vein of that "Tookishness" by virtue of physical heredity. If it weren't for Sam, we might be left thinking that someone had to have a bit of Elvish blood in order to be spiritually awakened. But as we leave Sam standing at the Havens with the sound of the Sea sinking into his heart, we know that it can happen to anyone - even us.
*Movie note: The movies seem to "get" Frodo and Sam's differing personalities, and even how they would affect the communication between the two. One example is the opening of LotR-TTT, with Frodo's "Nothing... just a dream." We can see Frodo's face (Sam can't), and know that the dream is anything but insignificant; the fact that it's "just" a dream has a different meaning for Frodo than it does for Sam (although in this instance, it seems that Frodo's aware of how Sam will interpret what he says, which isn't always true).
Without adding a second footnote for the next paragraph, I also appreciate the way the same movie handles Frodo's light as he lies asleep in Ithilien. Would the way the camera takes in the scene of Frodo basking in the sun, with the light shining on and around him, be the way someone without "the eyes to see" would have observed what Sam sees more deeply?
**This is an allusion I dropped in "unconsciously at first ," only later realizing that it's possible not everyone is familiar with it. It's from a poem by Francis Thompson (a poet Tolkien admired, BTW), speaking as someone who has spent his life running from God - Who never gives up pursuing him.