The entire Latin Noted Missal is now available online.
Latin Noted Missal
Latin Gloria Patri at the Officium (Introit)
Latin Sequentiarium
Indices

The Sarum Gradual Latin (3 vols), containing all the chants for the choir at mass is now available (2021) for purchase: see under ‘books‘.

The English Noted Missal is in preparation.
English Noted Missal
English Gloria Patri at the Officium (Introit).  (Palmer and Burgess, The Plainchant Gradual 1946/R 1965:302-304 includes the Sarum version of the Tones for the Gloria Patri in English, even though the proper chants of the mass are the Roman forms.)
While the above is in preparation, a Beta edition of the  Sarum English Performing Gradual is available as of July 2012.

Pearson, The Sarum Missal: ix-lv. gives an excellent, if in some respects now dated, historical introduction and explanation of the Sarum Mass.

Performance notes on the proper chants

Officium
The antiphon is intoned by one or more leaders, and taken up by the full choir.  On Sundays and feasts of saints with rulers of the choir, and within ruled octaves, and on masses of St. Mary and the Feast of the Place, the antiphon is repeated after the psalm-verse–before the ‘Gloria Patri’–and again after the ‘Gloria Patri’:
-Antiphon
-Psalm-verse
-Antiphon
-Gloria Patri
-Antiphon

When the choir is not ruled the first repetition of the antiphon is omitted.
-Antiphon
-Psalm-verse
-Gloria Patri
-Antiphon

During Passiontide the ‘Gloria Patri’ is omitted.
-Antiphon
-Psalm-verse
-Antiphon

I have yet to see Sarum rubrics indicating who is to sing the psalm-verse and the ‘Gloria Patri’.  The typical Roman practice today is for a leader to sing the psalm-verse up to the mid-point (colon or *), at which point the full choir continues; and likewise for a leader to sing the ‘Gloria Patri’ up to the end of ‘Spiritui Sancto’, at which point the full choir continues.  However, it is more typical in the Sarum practice for two (or four) soloists or rulers to sing verses.

There is no specific instruction for the timing of the commencement of the choral officium. Presumably it began with the vesting or with the completion of the vesting, if the vesting was in view, or with the beginning of the entrance of the ministers, and perhaps was signalled by the ringing of a bell, as is commonly done today. The choral officium continues during the prayers at the foot of the altar. The choral Kyrie follows immediately after the conclusion of the Officium.

Gradual
Normally one or more leaders intone the responsory after which the full choir commences the responsory again from the beginning and continues until the verse.   The leaders sing the verse; the full choir joins in for the final word(s) of the verse.  The leaders again intone the responsory, after which the full choir again commences the responsory from the beginning and concludes it.

On certain occasions the repetition of the responsory is omitted.  They are: double feasts (except the second mass of Christmas), the fifth and sixth days of Easter, the first Gradual on Wednesday and Saturday of the ‘four seasons’ (ember days) of Advent, Lent, and September, and generally when a Tract follows.

Exceptions to these principles are noted in the text.

On double feasts the soloists are three clerks in the pulpitum.
On Sundays and other ruled feasts the soloists are two boys in the pulpitum.
On unruled feasts with double invitatories the soloists are two boys at the Quire Step.
On unruled feasts with single invitatories the soloist is one boy at the Quire Step.

(If circumstances require a solo performance it would be appropriate to omit the repetitions of the incipit.)

Alleluya
Normally one or more leaders intone the Alleluya, after which the full choir commences the Alleluya again from the beginning and continues to the end of the neuma (jubilus). The leader(s) sing the verse; the full choir joins in for the final word(s) of the verse.  If a sequence shall follow, the leader(s) again intone the Alleluya. If there is no sequence, the leader(s) again intone the Alleluya, after which the full choir again commences the Alleluya from the beginning and concludes it with the neuma (jubilus).

On occasion, such as Saturday in Easter week and the feast of the Seven Brothers, there are two verses.  In these cases the choir joins in for the final word(s) of each verse; there is no repetition of Alleluya before the second verse.

On Wednesday and Saturday of the ‘four seasons’ (ember days) of Pentecost the first Alleluya is not repeated after the verse.

(If circumstances require a solo performance  it would be appropriate to omit the repetitions of the incipit.)

Sequence
There are no specific Sarum rubrics pertaining to the performance of the sequence. However, Newcastle University: Petre’s Gradual indicate through the use of bar lines the entry of the choir after the one or more words have been sung by the leader(s); barlines in the 1508 Gradual generally agree with Petre’s Gradual.  In the edition ‘*’ indicates this point.  Presumably the Choir side continues and completes the first line (double bar); from thence, alternation by sides would continue to the end.  It may be appropriate for the final line of the sequence to be sung by the full choir.  Amen is not sung at the end when the sequence is sung at mass.
The ‘Companion to the Missal’ suggests transpositions of the Alleluya and sequence so that the sequence may be sung directly after the Alleluya without a break or resetting of pitch.

Tract
The tract is sung by soloists or by the full choir.  Normally it is sung by four soloists which begin the first verse and conclude the final verse together, the remainder of the verses being sung alternately, two by two.  On Ash Wednesday, Friday after Ash Wednesday, The first Sunday in Lent, Mondays, Wednesdays and Fridays in Lent, including Holy Week, and Palm Sunday the tract is sung by the full choir in alternation.  On Easter Even the tract Laudate Dominum is sung by two.  Note, however, that  on Saturdays of the Ember Days there are also special tracts which contain refrains for full choir alternating with verses for soloists.

Offertory
It would appear that the offertory is intoned by one or more leaders and is continued and concluded by the full choir.

Offertory Verse
Offertory verses are sung on ferias in the seasons of Advent and Seputagesima through to Holy Week, and in the Mass for the Dead.  I have yet to see Sarum rubrics pertaining to the performance of the offertory verse.  It presumably would follow the pattern of the gradual, that is, sung by one or more leaders.  However, it is not clear whether the verse is concluded by the full choir or not, and it is not clear whether the responsory is repeated after the verse.  Given that the offertory verse is often tonally open at the end, and that no other musical item follows it directly (as the Alleluya may follow directly after a gradual verse, or a sequence may follow directly after a repetition of Alleluya–omitting the jubilus), it would make the most sense to repeat the whole responsory.  It will also be noted that the design of the concluding notes of the verse frequently seem to prepare for a repetition of the responsory from the beginning.  It may also be noted that the presence of offertory verses in the use of Sarum usually corresponds with masses that include a larger number of secrets (7 on the ferias of Lent, for example).  Thus the extended length of the offertories can be well understood.  (The Roman practice seems to be to repeat the latter part of the responsory as a repetenda, as is done in the Ott edition of the Offertoriale (1935), for example.)
Offertory verses do not appear in the earlier MS sources.  They are found in Cam-Queens-MS-28 (15th c), Petre’s Gradual (Newcastle) (c. 1370-1418), and the printed Sarum Graduals, 1508, 1527, 1532.  This may suggest that at some point in the 13th-14th c. existing offertory verses that had not been part of the Sarum Use were incorporated specifically as a feature of ferial masses in penitential seasons, a surprising hypothesis seeing that in other western uses the offertory verses fell permanently out of use.

Communion
It would appear that the communion is intoned by one or more leaders and is continued and concluded by the full choir.  The Sarum Rite does not include communion verses.

Performance notes on the ordinary chants

Kyrieleyson
Kyrieleyson is begun by the ruler(s) and continued by the choir, presumably in alternation.

Gloria in excelsis
Gloria in excelsis is intoned by the celebrant and taken up and concluded by the full choir without alternation.

Credo in unum Deum
Credo in unum Deum is intoned by the celebrant and taken up and concluded by the full choir without alternation.

Sanctus
Sanctus is begun by the ruler(s) and presumably continued by the full choir.  It may be effective to sing the iterations of Sanctus antiphonally thus: rulers, ‘other’ side, ‘choir’ side, and then continuing with full choir; Benedictus qui venit can be sung by soloists, with the full choir joining for the final Osanna in excelsis.  (Antiphonal performance echos Isaiah’s vision of the two seraphims (Is. 6:1.)

Agnus Dei
Agnus Dei is begun by the ruler(s) and continued by the full choir.  It may be effective if each iteration of Agnus Dei is begun by soloist(s), with the full choir joining in at ‘qui tollis peccata mundi’.

Ite missa est/Benedicamus Domino
This versicle ought sung by the Deacon.  If necessary, it may be appropriate to delegate this task to a cantor or soloist.  There appears to be no definitive evidence–except at the vigil of Easter and the vigil of Pentecost–that the response ‘Deo gratias’ is to be sung or said aloud, but it seems more likely that it was sung.  If it is sung, it should be sung by the full choir.