Shield specifically specifies that you are "hit" with an attack. Parry specifies that you are damaged.

Generally things that allow you to use your reaction will proc on one of several conditions:

• You are targeted. You take this action before you know the outcome of the roll
• You are hit. You take this action after you know the outcome of the roll
• You take damage. You take this action after you know the outcome, and after the damage has been rolled.

With each of these the circumstances of the player DM interaction is slightly different.

With the first, the DM says "The monster targets you with X" and you say "ok, wait, I'm going to use X" before he rolls the attack.

With the second, the DM says "The monster targets you with X, his roll is 20, does that hit your AC?", you say "It would, but I'm going to use Shield, so now it doesnt"

With the third, the DM says "The monster targets you with X, his roll is 20, that hits your AC, he deals 10 damage", and you say "I'm going to use Parry, and reduce the damage by 1d6".

With Shield, it is up to the DM whether or not he reveals the total of the roll or not. I generally think it is good practice to do so, but the DM is within his rights to ask for your ACs and keep track of the changes (this also creates more bookkeeping for the DM which can be a pain, just put your cards on the table).

It is likewise, up to the DM whether he tells you how much damage is done before Parry is applied. However, in general, I'd also prefer open communication here.

The result of a roll is not determined until the DM says what the result is, even in the case of a 1 or 20. There are cases, such as attacking an illusion or something immune to that damage type, wherein the DM lets the player roll just to hear the dice clatter.