I have never fully understood this. There is a reverend mother with the Fremen, and she seems to be able to talk with Jessica as though she is Bene Gesserit. However, it is also implied that she is purely Fremen or has at least gone native. What was her affiliation? Was she a BG missionary/sleeper agent sent to the Fremen to one day help a BG in need? Was she Fremen through and through? Something else?
She was just a Fremen
"...the Missionaria Protectiva has been in there and softened it up somewhat"
Rev Mother Helen
"With the Lady Jessica and Arrakis, the Bene Gesserit system of sowing implant-legends through the Missionaria Protectiva came to its full fruition. The wisdom of seeding the universe with a prophecy pattern for the protection of B.G. personnel has long been appreciated..."
Princess Irulan 'narration'
"So, the Missionaria Protectiva planted religious safety valves all through this hell hole. Ah well...it'll help."
"Our Missionaria Protectiva seldom fails. A place was prepared for us in this wilderness. The prayer of the salat has carved out our hiding place. Now I must play the part....to rogue peoples who've been so heavily imprinted with our Bene Gesserit soothsay that they even call their chief priestesses Reverend Mothers."
These quotes indicate that the Fremen have been highly influenced by very old thought-seeds planted by the Bene Gesserit MP, not that the Fremen have an actual Bene Gesserit amongst them when Jessica arrives. It does seem though, that, given their access to spice and the water of life, that the Fremen priestesses to have some ability to see into the same places that the BG do when it comes to prescience and so she is as close to a 'real' Reverend Mother than might be expected at first.
(note: the above answer is based soley on the first novel. Valorum's answer has an upvote from me because it references canon material [though not explicitly written by Frank Herbert]).
Sister Ramallo was, we learn in Dune: House Harkonnen, part of a delegation of Bene Gesserit sisters sent to Arrakis to plant a favourable religion and to prepare the way for their Kwisatz Haderach, should he (or any of the sisterhood) need an emergency support structure on the planet.
She evidently decided to 'go native' and remain amongst the Fremen when her sisters left, but there's no indication that her overall allegiance wasn't aligned to the BG.
Slowly, like a dreamer dancing inside her own consciousness, Margot felt herself return to the stone-walled chamber, with the drug-induced vision only a flickering memory. Fremen continued to touch their fingers to the hanging droplets, tasting, moving to the side so that others could partake. Euphoria spread like dawnlight in the cavern.
“Yes, once I was a Reverend Mother,” Ramallo told her, at long last. “Many years ago I knew your Mother Superior.” Still fogged by reverberations of the powerful drug, Margot couldn’t even act shocked, and the old woman nodded. “Sister Harishka and I were classmates".
This is also stressed in the Dune TV mini-series, in which Jessica recognises and refers to her as a BG 'missionary'
Jessica: She came from the southern sietch.
Paul: Why? To see me.
Jessica: To see you. She's Bene Gesserit. One of our missionaries.
Interestingly, the Dune Encyclopaedia article on Ramallo takes a slightly different slant, suggesting that the Bene Gesserit had spent considerable time forging links with the wild Reverend Mothers of Arrakis, indoctrinating them into Bene Gesserit in all but name, and that Ramallo may have been part of a group of young Sayadinas-in-training that were visited by, among others, Helen Mohiam.
On such a trip to Dune, Helen accompanied her, and R.M. Croesia records in her Memoirs that her student made a close friend of a Fremen girl named Alynda—and Alynda was Ramallo's first name. If this was Ramallo, could she have known; when she befriended an only slightly younger girl named Helen Mohiam, that their lives would be inextricably twined? Could either of them foresee then that the secret child of one would become the spiritual child and heir of the other?
As I understood and remember it, the BG have send their The Missionaria Protectiva to all corners of the universe to prepare for their plans or conveniences.
"Jessica thought about the prophecy—the Shari-a and all the panoplia propheticus, a Bene Gesserit of the Missionaria Protectiva dropped here long centuries ago—long dead, no doubt, but her purpose accomplished"
The 'no doubt' addition suggests this has not been confirmed. My interpretation was that the party sent to Arrakis at some point lost contact with the rest of the sisterhood, possible according to plan. It turns out they did turn Fremen, but never completely stopped being Bene Gesserit. Since the reverend mothers of the Fremen underwent the same Spice Agony and thus the same change as the 'mainstream' Bene Gesserit, they could look into their female ancestors mind, and thus they had a direct line of communication with the (deceased) The Missionaria Protectiva sent to Dune in the first place.
The assumption that Fremen Reverend Mother is the same as Bene Gesserit title may be incorrect. There is similarity of the title, rites and the results of it on Fremen Sayyadina and B.G. Sister. There are multiple mentions of Missionaria Protectiva and that it apparently visited Dune long time ago to plant seeds of support for any B.G. who may end up stranded there, but nothing is confirmed, except maybe for the title only:
REVEREND MOTHER: originally, a proctor of the Bene Gesserit, one who has transformed an “illuminating poison” within her body, raising herself to a higher state of awareness. Title adopted by Fremen for their own religious leaders who accomplished a similar “illumination.” (See also Bene Gesserit and Water of Life.)
So far, the strongest argument AGAINST the similarities being copied from B.G. is in the description of Jessica's experiences when she underwent the Rite that saw her becoming Reverend Mother. Again, there are also mentions of similarities between the Fremen rite and B.G. rite, but then again - process forces the form, so this may by another coincidence:
There was familiarity to the rite, and Jessica looked back at the Reverend Mother Ramallo. The old woman’s eyes were closed and she sat hunched over as though asleep.
And she knew with a generalized awareness that she had become, in truth, precisely what was meant by a Bene Gesserit Reverend Mother. The poison drug had transformed her
And she saw the thread of the past carried by Sayyadina after Sayyadina—first by word of mouth, hidden in the sand chanteys, then refined through their own Reverend Mothers with the discovery of the poison drug on Rossak … and now developed to subtle strength on Arrakis in the discovery of the Water of Life
Last quote is very indicative, because it literally says Fremen had Reverend Mothers BEFORE arriving on Arrakis, so if anything, Missionaria Protectiva would just reinforce most of the beliefs and superstitions of the Fremen, but not inject them. I am trying to nail down when Rossak's discovery happened, but it is quite, quite old one.
Of course, anything here is limited to Original Dune, as I don't really accept anything after Chapterhouse as canon.
[Not because I'm some kind of purist, but simply because everything not written by Frank Herbert to be so close to horsec**p that it's almost indistinguishable. Pity, it had so much potential.]