So I spawned in creative, no mods installed, barely modified terrain and placed down a double chest. They both have separate inventories, and one side doesn't make noise, nor does it play the chest open animation.
Here's a video demonstration: https://www.youtube.com/watch?v=Rnpselp9AIA&feature=youtu.be
Environment
Windows 10, Java 8
Cannot reproduce, this works fine for me.