Q.: Has anyone written on NLM about the rubrics for a private Mass? Does the priest say all of the dialogue parts, or forego them? Are some still said, such as the dialogue before the Preface, but others, such as the "Orate fratres," not said?
Before addressing the "how to," we need to be clear about terminology and about the mind of the Church concerning such celebrations of the Eucharist. First, the term "private Mass" does not refer to Mass with neither a server nor a congregation, but rather to an unscheduled Mass, regardless of the number of faithful present and regardless of the degree of solemnity (a High Mass can still be a private Mass). What the enquirer has in mind is a Missa solitaria, Mass at which no one assists.
Second, such Masses are generally discouraged, and Church synods have actually forbidden them. J.B. O'Connell's The Celebration of Mass, 4th ed. (Milwaukee: Bruce Publishing, 1964) gives the following directive (pp. 365-66):
Only in case of real necessity may a priest say Mass when there is no one who can and may serve or who can at least reply, e.g.: (a) to consecrate a Host to give Viaticum, (b) to say Mass on a Sunday or holyday of obligation so that others -- or even he himself -- may fulfill the duty of assisting at Mass ... If only an untrained server be, on occasion, available, he should be used, since he will be capable, at least, of carrying out some of the chief ceremonies, e.g., handing the cruets, transferring the Missal, ringing the bell.
Canon 906 of the 1983 (current) Code of Canon Law, which corresponds to canon 813 of the 1917 Code, states: "A priest may not celebrate the Eucharistic Sacrifice without the participation of at least one of the faithful, unless there is a good and reasonable cause for doing so." This needs to be read in the light of canon 904, which earnestly recommends that priests celebrate Mass daily; this strong recommendation would itself seem to provide a "just and reasonable cause," provided that the priest make an effort to obtain the assistance of at least one member of the faithful.
So much for preliminaries. Now, to answer the question:
O'Connell explains (p. 366) that when celebrating Mass without a server (whether or not other members of the faithful are present), the priest says all of the responses himself, but (a) he says the Confiteor once only, omitting the words "vobis, fratres" and "vos, fratres" and saying at the end "Misereatur nostri"; and (b) in the reply to "Orate, fratres," he says "de manibus meis" instead of "de manibus tuis" (which makes perfectly good sense). Furthermore, in moving the Missal, the priest does not genuflect when passing the tabernacle but bows his head to the cross. It would be convenient before Mass to pour water into the finger bowl so that the priest can dip his fingers at the Lavabo. A footnote suggests that, if no one is present, "it does not seem appropriate to turn around and address Dominus vobiscum to nobody and absurd to give the blessing in this case."
Of course, these rubrics apply to the Missal of 1962. So, there was some provision made in the rubrics for the rite of the celebration of Mass without a server. But neither the 1962 Missal nor any of the "typical" editions of the Missal of Paul VI (1970, 1975, 2002) envision Mass with no one present but the priest. In the part of the Missal containing the Order of Mass, the section titled Ordo Missae sine populo in the 1970/75 editions, now called Ordo Missae, cuius unus tantum minister participat in the 2002 edition, presuppose one server. Yet, as was noted, canon law does not absolutely forbid Missae solitariae. Therefore, in a spirit of continuity with the preceding practice, I would apply the same rules laid out in O'Connell, mutatis mutandis, to the celebration of a Missa solitaria in the ordinary form. Perhaps (though I doubt it) the Church will remedy the lack of precise instructions for Masses of this sort.