The prayers of the faithful seem to be in that transitional stage between the Word and the Eucharist, but the GILM places them with the Word of God, and here’s why:
30. In the light of God’s word and in a sense in response to it, the congregation of the faithful prays in the universal prayer as a rule for the needs of the universal Church and the local community, for the salvation of the world and those oppressed by any burden, and for special categories of people.
The celebrant introduces the prayer; a deacon, another minister, or some of the faithful may propose intentions that are short and phrased with a measure of freedom. In these petitions “the people, exercising its priestly function, makes intercession for all men and women,”  with the result that, as the liturgy of the word has its full effects in the faithful, they are better prepared to proceed to the liturgy of the Eucharist.
There is a certain logic that after the Word is broken open through proclamation and preaching, and that the Christian faith is affirmed in the Creed, that the people are ready to act as a “royal priesthood.” This is probably the best argument I’ve seen for a presider to remain silent during these prayers, ensuring they are literally from the people. To this end, my parish has had a long tradition of assigning to parishioners the composition of these prayers for Sundays and holy days, and liberally opening up this moment of daily Mass for their petitions.
Some of my colleagues fuss a bit about announcing these prayers from the ambo, but like the GILM, I don’t have a problem with it:
31. For the prayer of the faithful the celebrant presides at the chair and the intentions are announced at the ambo. 
The assembled congregation takes part in the prayer of the faithful while standing and by saying or singing a common response after each intention or by silent prayer. 
I’m sure you all know that singing a common response is acceptable. Did you know that silent prayer after each response is also a possibility. Would a well-trained assembly respond automatically? If you were ever to implement silence after each petition, the person rendering these would need to be prepared, as would the people in the pew, most likely. My own sense would be to implement a sung response for some liturgical seasons and feasts. Perhaps the silent response might be better incorporated into a remembrance of the deceased, of ill family members or friends, or of the more common “intentions held in our hearts.”
Thoughts or suggestions?