In what manner did Peter, James and John appear to Joseph Smith?

According to Robert T Beckstead The manner may have been by means of psychedelic induced visions. The introduction to an abstract delving the use of psychedelic mushroom laced alcohols says:

Did hallucinogens facilitate Joseph Smith’s visions & those of early Mormon converts? In his 1975 book, Hearts Made Glad, Lamar Petersen carefully documented the use of intoxicants by Joseph Smith and early converts to the LDS Church. While mostly interested in the consumption of various fermented and distilled alcohols, Petersen also noted strange behaviors associated with the sacramental use of what seemed to outside observers to be medicated wine. It appears that soon after the Church was organized in New York and later in Ohio, members partook of wine in sacrament meetings which occasioned visionary states and strange behaviors not typically associated with alcohol consumption or intoxication. It is my thesis that beginning at a young age, Joseph Smith experimented with psychedelic plants and that many of Joseph Smith revelations and much of his behavior can be attributed to the use of psychedelics. Following Joseph Smith’s death, the pragmatic Brigham Young had no interest in psychedelic material, or was unaware of its use, and hence it did not become a part of Utah Mormonism. However, James Strang and Fredrick M. Smith (Joseph Smith’s grandson and president of the RLDS Church) perpetuated the use of psychedelics in their branches of Joseph Smith’s original movement. The use of psychedelics by the Strangites and the RLDS Church could not be sustained.

The full text of the abstract is found http://www.mormonthink.com/files/restoration-sacred-mushroom.pdf

Both John the Beloved and John the Baptist presently have physical bodies, as does Peter, James, Jesus Christ, Moses, Elias, and Elijah (ect).

Some of these men were resurrected (after Jesus Christ arose and ascended to heaven) and some of these men were temporarily translated (not tasting of death to continue their work here in mortality) but will become resurrected beings when Jesus Christ Comes Again to gather them (the Second time). Either way these men have physical bodies designed to accomplish their given assigned tasks.

As far as doctrine, usually the details of such events aren't explained. Members of the Church usually rely on faith and trust that, because God is all-powerful, He can bring to pass miraculous events in whichever way He may choose.

That being said, there is scriptural evidence that can help us to infer what may have happened. For example, it isn't out of the question for God to assist mortal or immortal persons regardless of resurrection. For example, consider this passage from when Nephi in the Book of Helaman was teaching:

15 And it came to pass that when Nephi had declared unto them the word, behold, they did still harden their hearts and would not hearken unto his words; therefore they did revile against him, and did seek to lay their hands upon him that they might cast him into prison.

16 But behold, the power of God was with him, and they could not take him to cast him into prison, for he was taken by the Spirit and conveyed away out of the midst of them.

Helaman 10:15-16

Here Nephi, a mortal man, was "conveyed" by the power of God. It is likely that God used a similar method to reunite John with Peter and James for the restoration of the Melchizedek Priesthood.

A similar event documents the travel of Moroni after appearing to Joseph Smith Jr. as a resurrected being:

After this communication, I saw the light in the room begin to gather immediately around the person of him who had been speaking to me, and it continued to do so until the room was again left dark, except just around him; when, instantly I saw, as it were, a conduit open right up into heaven, and he ascended till he entirely disappeared, and the room was left as it had been before this heavenly light had made its appearance.

The Testimony of Joseph Smith

This gives us a glimpse of what the travels of Peter and James may have been like. Peter and James at this time were presumably resurrected beings, as they were participating in physical ordinances. The Church teaches that the dead cannot perform their own ordinances such as baptism, confirmation, and the confirmation of Priesthood Authority. To this end, living members perform ordinances on behalf of the dead. Therefore, Peter and James must be living (resurrected) in order to participate in the ordinance of confirmation of the Melchizedek Priesthood.

Peter and James, we infer, have been resurrected already and have physical bodies. The laying on of hands was performed by physical beings to physical beings.

A few references:

• Peter, James, and Moroni have been resurrected (see Supporting Statement B)

• John the Baptist is also apparently resurrected

According to Author: McConkie, Mark L.

Latter-day Saint scriptures speak of a unique class of beings, persons whom the Lord has "translated" or changed from a mortal state to one in which they are temporarily not subject to death, and in which they experience neither pain nor sorrow except for the sins of the world. Such beings appear to have much greater power than mortals. All translated beings will eventually experience physical death and resurrection (MD, p. 807-808). Translation is a necessary condition in special instances to further the work of the Lord.

Translated beings are not resurrected beings, though all translated beings either have since been or yet will be resurrected or "changed in the twinkling of an eye" to a resurrected state (3 Ne. 28:8). In effect, this last change is their death, and they therefore receive what amounts to an instantaneous death and resurrection. Resurrection is a step beyond translation, and persons translated prior to the resurrection of Christ were resurrected with him (cf. D&C 133:54-55); it is expected that those translated since Christ's resurrection will be resurrected at his second coming.

During the period from Adam to Melchizedek, many faithful persons were translated. Enoch and the righteous residents of his city of Zion were translated not many years after Adam's death (Moses 7:18-21, 31, 63, 69; D&C 38:4;45:11-14;84:99-100; Gen. 5:22-24; Heb. 11:5). During the period from Enoch to Noah, it appears that faithful members of the Church were translated, for "the Holy Ghost fell on many, and they were caught up by the powers of heaven into Zion" (Moses 7:27).

After the Flood, others were also translated. In his inspired rendition of the Bible, Joseph Smith tells of many who "were translated and taken up into heaven" (JST Gen. 14:32-34). Fewer translations apparently occurred in the New Testament era, though John the beloved (John 21:20-23; D&C 7) and the three Nephites were translated (3 Ne. 28).

Translated beings are assigned special ministries, some to remain among mortals, as seems to be the case of John and the Three Nephites, or for other purposes, as in the case of Moses and Elijah, who were translated in order to appear with physical bodies hundreds of years later on the mount oftransfiguration prior to the resurrection of Christ. Had they been spirits only, they could not have laid hands on the mortal Peter, James, and John (cf. D&C 129:3-8). Why those of Enoch's city were translated, we are not specifically informed, although the Prophet Joseph Smith explained the role of translated beings thus: "Many have supposed that the doctrine of translation was a doctrine whereby men were taken immediately into the presence of God, and into an eternal fullness, but this is a mistaken idea. Their place of habitation is that of the terrestrial order, and a place prepared for such characters He held in reserve to be ministering angels unto many planets, and who as yet have not entered into so great a fullness as those who are resurrected from the dead" (TPJS, p. 170).

The scriptures do not define differences between transfiguration and translation, but it appears that transfiguration is more temporary, as in Matthew 17:1-9 andMoses 1:11, occurring primarily to permit one to behold spiritual things not possible in the mortal condition.