White is white. If something is white, it doesn't absorb visible light. If it did, it would not look white. Bringing in white paper in place of the fabric backdrop should not make any difference.
If it's coming out grey in the photos, you just don't have the ratio of subject to backdrop lighting correct. You should be lighting the backdrop 1 to 2 stops brighter than the subject. How much the backdrop lights spill side lighting onto your subject depends on: the ratio, the distance between the subject and backdrop, and what additional structures you have in place to flag off the backdrop lights (such as v-flats, or the use of gridded softboxes to light the backdrop). For a headshot, you probably don't want the backdrop lights hitting your subject at all, so flag them off or put distance between the subject and backdrop. When the backdrop lights sidelight your subject, you get more of a high-key look, like a cosmetics ad beauty shot, not a clear separation from the backdrop as in your profile pic.