Tested with most recent versions on 1.21.1 as of writing - Core 1.3.47.999, Storage 1.4.37.1162, with NeoForge 21.1.138 (and double checked in 21.1.173).

When a double chest is placed, only its left half will give off comparator output. This appears to be independent of the orientation in the world and of whether the left or the right side is placed first.

Crimson chests here were placed left, then right. Warped chests were placed right, then left. All chests in this screenshot have one stack, in the first inventory slot.

It's the same if a slot in the second half of the inventory is filled, so it doesn't seem to be something like half the inventory corresponds to one block space, and the other half to the other block space.

And it's the same when the comparator reads from a different side.

For comparison, Create's threshold switch can read the contents accurately from either block space.

I've noticed the Storage Tool will also only link with the left half of a double chest (although can still see and interact with the full double chest just fine).