In the USA very broad authorisation has been given.
From the 2002 General Instruction of the Roman Missal (GIRM) approved for the USA, which can be accessed from
romanrite.com/girm.html :
“61. … The responsorial Psalm should correspond to each reading and should, as a rule, be taken from the Lectionary.
It is preferable that the responsorial Psalm be sung, at least as far as the people’s response is concerned. Hence, the psalmist, or the cantor of the Psalm, sings the verses of the Psalm from the ambo or another suitable place. The entire congregation remains seated and listens but, as a rule, takes part by singing the response,
except when the Psalm is sung straight through without a response. In order, however, that the people may be able to sing the Psalm response more readily, texts of some responses and Psalms have been chosen for the various seasons of the year or for the various categories of Saints. These may be used in place of the text corresponding to the reading whenever the Psalm is sung. If the Psalm cannot be sung, then it should be recited in such a way that it is particularly suited to fostering meditation on the word of God.
In the dioceses of the United States of America, the following may also be sung in place of the Psalm assigned in the Lectionary for Mass: either the proper or seasonal antiphon and Psalm from the Lectionary, as found either in the Roman Gradual or Simple Gradual or in another musical setting;
or an antiphon and Psalm from another collection of the psalms and antiphons, including psalms arranged in metrical form, providing that they have been approved by the United States Conference of Catholic Bishops or the Diocesan Bishop. Songs or hymns may not be used in place of the responsorial Psalm.”
So it seems to me that in the USA a bishop can make a collection with any psalm and any translation of the psalm for a given day.
In Australia, for example, the bishop has no such authority, in its GIRM n. 61: “The following may also be sung in place of the Psalm assigned in the Lectionary: either the responsorial gradual from the Graduale Romanum, or the reponsorial Psalm or the Alleluia Psalm from the Graduale Simplex, in the form described in these books.”
