Q: Are practitioners of Nichiren Daishonin's Buddhism directly related to him, as the SGI teaches? Or are they related to him only through their relationships with ordained clergy, as Nichiren Shoshu teaches? 

Since being forced out of Nichiren Shoshu, the Soka Gakkai has been teaching that its members are the direct disciples of Nichiren, and that they have a "direct connection" to him. 

There are two levels to this direct connection: 

  1. The SGI has published the Gosho, and translated most of it onto other languages. Because of that, we are able to read exactly what Nichiren taught, and take his very words as our doctrine and guidance. 
  2. On a more profound level, through our faith and practice, we "commune" directly with the Person and Law, the fundamental nature of life, the Mystic Law. Everything needed to attain enlightenment is to be found in the practitioner's own life, including his/her actions as a bodhisattva through SGI activities. There is no need for an intermediary.
Naturally, this idea gives Nichiren Shoshu fits, as it precludes the necessity of priests, their blessings, and ceremonies.

However, theirs is the more traditional view, and much easier to understand. Because he spent a lifetime refuting the established, traditional views of "connections", Nichiren himself suffered persecution throughout his life — overturning deeply entrenched thinking is a difficult task.

The traditional view, in this case, is that the entirety of the Buddha's enlightenment has been infused into a single Gohonzon, the Dai Gohonzon enshrined at Taisekiji; and has been transferred from Nichiren to Nikko to all 67 High Priests of Nichiren Shoshu, who are then capable of transferring it from the Dai Gohonzon into the Gohonzons given to individuals; it is the transference by the High Priest that makes those Gohonzon efficacious. In other words, without the High Priest of Nichiren Shoshu, no one in the world can ever attain enlightenment.

But Nichiren himself scoffed at the idea of a "special transmission." For instance, in 
"The Opening of the Eyes," he writes:

Are the voice-hearers protecting those who disparage the Lotus Sutra as a mere written teaching and who put forth their great lies about what they call a special transmission? —Writings of Nichiren Daishonin (WND), p. 247
Since the priesthood sees "direct connection" as at the core of what they call heresy, it is worth the time here to address the topic.

From the priesthood's 100 Questions and Answers

31. Is the "Direct connection to the Daishonin" propounded by the Gakkai correct? It is not correct. The true intention of the "direct connection to the Daishonin" is to deny the bequeathal of the lifeblood of the successive High Priests as well as the seven hundred year traditions of the sect, and to justify their independence as the Ikeda Religion. 
Of course it is correct because the ideas of "bequeathal" solely to priests and the "700 year traditions" of Nichiren Shoshu are deviations from what the Daishonin taught — which, by the way, includes direct connection. For instance:
"Shakyamuni Buddha who attained enlightenment countless kalpas ago, the Lotus Sutra that leads all people to Buddhahood, and we ordinary human beings are in no way different or separate from each other. To chant Myoho-renge-kyo with this realization is to inherit the ultimate law of life and death." —WND, p. 216
"Since every single word constitutes the soul of the Buddha, Shakyamuni Thus Come One will protect persons who practice this sutra as though he were protecting his very own eyes. He will accompany them just as a shadow accompanies a body." —WND, p. 339
"The Buddha who is the entity of Myoho-renge-kyo, of the Life Span Chapter of the essential teachings, who is both inhabiting subject and inhabiting realm, life and environment, body and mind, entity and function, the Buddha eternally endowed with the three bodies — he is to be found in the disciples and lay believers of Nichiren. Such persons embody the true entity of Myoho-renge-kyo." —WND, p. 420
"Question 31" continues:
It is always important that we have faith and follow the guidance of the High Priest of the time. The reason is because the Daishonin's innermost enlightenment was bequeathed to only Nikko Shonin, and afterwards it was bequeathed to the successive High Priests from Nichimoku Shonin on down up until today, as Nichiren Daishonin stated in the "Minobuzan Fushoku Sho" —"Mt. Minobu Transfer Document," also known as "Ikegami Transfer Document" 

"The fifty year teachings of Shakyamuni are bequeathed to Byakuren Ajari Nikko. He is to be the steward of Minobuzan Kuon-ji. Lay believers and priests who defy this belong to the slanderous multitude" —Nichiren Daishonin Gosho Zenshu [in English: The Complete Writings of NichirenDaishonin], p. 1600, Shinpen, p. 1675

Nichiren Shoshu has deliberately changed "the teachings are bequeathed" to "the enlightened life of the Daishonin is bequeathed." There is no magic in the Daishonin's Buddhism — one cannot "bequeath" enlightenment from one person to another, in the sense of performing a ceremony which causes that person assume the Buddha's enlightenment. In the Daishonin's Buddhism, the only way to enlightenment is faith in the Gohonzon, and practice for kosen-rufu, even in the face of great obstacles.

In fact, the Daishonin, in appointing Nikko, said Nikko "should be the leader of propagation". Likewise, Nikko, in his 26 admonitions (#13): 

"Until kosen-rufu is achieved, propagate the Law to the full extent of your ability, without begrudging your life."
Clearly, what they "bequeathed" to their closest disciple was responsibility, not authority or privilege. Over 700 years, Nichiren Shoshu has attempted to justify its existence — which it could not do by means of "propagate without begrudging your life", by introducing the concepts of absolute authority of the high priest, and the need for faith in the high priest. Of course, these ideas are direct contradictions of Nikko Shonin's Article 17, which gives believers the responsibility to protect the teachings, and to correct the high priest if he deviates from them. In writing this article, Nikko Shonin clearly indicates that a high priest can be mistaken, that absolute authority should not lie with a single person.

"Question 31" continues:

Priests and lay believers who defy the bequeathal of the lifeblood received by only a single person "belong to the slanderous multitude," and are committing a tremendous slander. 
Only if one buys into the Nichiren Shoshu distortions, which true disciples of the Daishonin do not do.

"Question 31" concludes:

Nikko Shonin as well admonished Sado Island believers, saying, "As I expected, many of the disciples who appeared even after the Sage's passing are saying that they are the Sage's direct disciples. These people are slanderers." —Nichiren Shoshu Rekidai Hossu Zenshu [in English: The Complete Writings of the Successive High Priests of Nichiren Shoshu], Vol. 1, p. 184 
The writings of the various high priests, we're told, contains lots of "rumors" — or so the temple says whenever the SGI quotes something from one of them. But in its hypocritical worldview this is a report of events, and not just some wild rumor.

Be that as it may, this doesn't contradict the Gosho — it is obviously a different context from what is meant by "direct connection". Immediately after the Daishonin's passing, many people may have come forward to claim they had been taught directly by him while he lived in Japan. This is obviously different that what is meant by "the Buddha, the Law and we ordinary human beings are in no way different or separate from each other."