If you place only one hopper underneath a double trapped chest, that hopper will only grab items from either the top half (top 3 rows) or bottom half (bottom 3 rows) of the chest, depending on the orientation of the setup (Hoppers on the north or west half of the double chest will take from the top 3 rows. Hoppers on the south or east half of the double chest will take from the bottom 3 rows).
With normal chests, a single hopper can grab items from an entire double chest.