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)
What is the problem? The player cannot open the 'lid' because it is blocked from above, but the hopper can still place items in (from the block above the other side of the chest). What's bad or incorrect about that?