When one picks up a coin, there is a connection made between mind of the holder and the entity in the coin and an imprint (a basic copy) of the Fallen is placed in his mind.
Also, there has to be constant contact in order for the holder of the coin to tap to full power and abilities of the Fallen. That is why every KoBD places the coin in direct contact (including, for example placing it under skin or swallowing it constantly).
Harry picked up the coin and then refused to take it up ever again. So all he had was the imprint. But... To answer the OP's question another one must be asked:
Who is Lasciel?
She let out a cold little laugh. "Such arrogance. Do you think you could change the eternal, mortal? I was brought to life by the Word of the Almighty himself, for a purpose so complex and fundamental that you could not begin to comprehend it. You are nothing, mortal. You are a flickering spark. You will be here, and be gone, and in the aeons that come after, when your very kind have dwindled and perished, you will be but one of uncounted legions of those whom I have seduced and destroyed." Her eyes narrowed. "You. Cannot. Change. Me."
So, taking above into account it stands to reason that if there was any connection between the two, the below would not be possible:
I nodded agreeably. "You're right. I can't change Lasciel. But I couldn't prevent Lasciel from walking out of the room, either." I eyed her hard and lowered my voice. "Lady, you ain't Lasciel."
"You're an image of her," I continued. "A copy. A footprint. But you've got to be at least as mutable as the material the impression was made upon. As mutable as me. And hey, I've got newfound anger issues. What have you got that's new?"
White Night, chapter 33
So, to use the analogy from the OP: Fallen knows fact A, thus Shadow knows it, too. But reverse flow is happening only when holding the coin.
Again, White Night, chapter 41:
"I can't," she replied, her voice anguished. "She would never forgive that. Never accept me back into her... just take the coin. Harry, just take the coin. P-please."
It can be reiterated that with any connection Fallen-Shadow above would not be possible.