In singleplayer at least, in all gamemodes, and on xp and seven, a hopper placed on the unblocked half of a trapped chest can place items in it, but not on the blocked half or with a normal blocked double chest.

They are able to take items from the unblocked half as well.

This is a consistency problem because it's completely impossible to use a hopper in any way with a normal chest in the same situation.

(By blocked chest, I mean a chest with a block directly above it)