Ever notice how a good organist will accompany a cantor with just the upper manuals on a light registration for the first part of a responsorial psalm, then crank up the volume, adding the pedals and a much fuller registration when the assembly is supposed to join in? Even if the cantor didn't raise his/her arms to invite the assembly to sing, they'd feel invited to sing simply by the fact that music changed and is suddenly full and supporting them. Our contemporary ensembles (bands) should do the same.
Here's a video that shows some examples of how we did this sort of thing this past Sunday.
These aren't the only ways to do these songs, by any means. We had only guitar, piano, bass guitar, drums, and 3 singers. More instruments give you more options to be creative and orchestrate the music.
But whatever you choose to do, however you choose to play the music...make it inviting to the assembly.