It varies, I think. In the Office psalms always end with a doxology. Both Graduals indicate it for Entrance* and Communion, but are silent about it for Gradual/Responsorial Psalm, and for Offertory. *in GR can be omitted if singing it would delay the action of the Mass. AFAIK it has never appeared at the Responsorial psalm in Missal or Lectionary.
Gelineau started his psalm project in 1953, and the Grail psalms are copyright 1963, so both precede any SC changes.
GS is fairly clear that for Entrance you can make any selection from the verses listed (provided it is coherent) and then end with Gloria ... .

There appears, historically, to have been a lot a of variation in how responsorial singing was handled, in general. This variation includes the inclusion or omission or half-omission ('half-doxology') of the Gloria Patri. Variation is especially wide in the Divine Office but I believe has, historically, extended to the mass propers as well.

I hasten to add that I don't know anything about the history of the modern Responsorial Psalm specifically, however.

dad29, no not clearly, but it is implied by GR Praenotanda #17

... Cantus autem eodem modo peragitur ac cantus ad introitum, ...

Which is same text as Ordo cantus Missae #17 DOL 535 says published 1972, promulgated 1974 ??

De Ritibus Servandis in Cantu Missae (1961)
IX. After the Communion of the priest, the full choir sings the Antiphon which is thus named, the Intonation being sung by one, two or four cantors as in the case of the Introit. When there are other communicants, the Antiphon is begun when
the priest distributes Communion. If the Antiphon is taken from a Psalm, other Verses of the same Psalm may be sung. In that case the Antiphon may be repeated after every Verse or two Verses; and when the Communion is ended, Gloria Patri followed by the Antiphon is sung. ...

I do not pretend to be speaking definitively, but I was once told that Gloria Patri was not said with psalms within the mass, only the office. Clearly, the Introit, in the form in which it lingers yet, is a disputation of this. Too, it seems to be current practice to sing Glory be... after psalms during the Communions.

Chris -

It would seem to me that The Introit, including, as it does, an antiphon and a verse with its Gloria Patri, would be sung entire whenever it is sung. The GIRM, after all, does call for the propers as they appear in the GR. This would mean that the Antiphon with its verse and Glory be..., as they appear in the GR, are expected to be sung at the Novus Ordo.

It is my opinion (and only my opinion!) that if omitting the verse and Glory be... and singing only the antiphon would get The Introit 'in the door' where 'time' or other such excuses are unduly significant factors, then one might consider doing this.

When the Gloria Patri can be added to the Entrance Chant or Communion Chant is explained in the Introduction to the Ordo Cantus Missae:

In this manner, the alternation of the antiphon and verses can be as long as is necessary to accompany the procession. However, before the antiphon is repeated at the end, Gloria Patri, Sicut erat, can be sung as a final verse, in the same manner as one of the verses already sung. If the Gloria Patri has a particular melodic ending, that same ending is to be used in all the verses.

If it happens that the chant, without the verse Gloria Patri and the repetition of the antiphon, becomes too protracted, the doxology is omitted. If the procession is shorter, only one verse of the psalm is used, or even the antiphon alone, with no verses added. (no. 1)

When the priest receives the Body of the Lord, the Communion antiphon is begun. The chant is carried out in the same manner as the Introit chant, and in such a way the cantors also may receive the sacrament. (no. 17)

No Gloria Patri is added tot the Gradual, Alleluia or Offertory Chant, because of their responsorial character.

I believe I've seen it claimed that the Gloria Patri was introduced as the final verse of every psalm in antiphonal singing of the psalms as introduced by Ignatius of Antioch.

It took its present form in response to Arianism -- apparently the Arians preferred (for obvious reasons) a version that used 'per' instead of 'et' after 'Patri'.

For reasons that I do not understand (but maybe someone here does?) as the various chants associated with various parts of the mass evolved, verses were dropped, especially from graduals, alleluias, and offertories. I do not believe (but am certainly prepared to be corrected) that this evolution had anything to do with their being 'responsory' in character -- indeed, in some cases the loss of verses seems to occur in parallel with the loss of the responsory character of the chant.

I can appreciate why the Introit and Communion chants would have verses and why -- as has been the custom for many centuries -- one would use more or fewer of those verses as needed in the moment. (There are references in even pre-Trent missals of the priest giving a signal to the choir to skip to the Gloria Patri in the Introit.) But the Offertory is also an accompaniment to a procession, and yet it appears that the verses disappeared from it rather early. I wonder why?

Maybe somebody who knows about the history of the liturgy can chime in.

In some versions of the Divine Office, there is responsorial singing that includes the Gloria Patri. The practice appears to be quite ancient. For example, Amalarius' De Ordine Antiphonarii (9th century) describes the use of the Gloria Patri in responsorial singing in the Roman Office. But clearly there was great variation in practice -- in fact, Amalarius begins his Antiphonary by mentioning the recent discovery of many antiphonaries in the library of Corbie, and the fact that they all disagree with one another.

Introit and Communion final verses are Gloria Patri at Mass.

Psalmody in the office ends with a Gloria Patri before the antiphon is repeated.

Historically, the offertory is a responsory, not an antiphon, since its verses are melismatic, not psalmodic, for which no Gloria Patri is foreseen. That old hand-missals and even the 1962 Liber refer to the offertory as an antiphon is only evidence that the priority of the traditional chant has been obscured.