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.
RAW
No damage dice => No dice to double
Sucks, but them's the rules. So a critical hit with an unarmed strike does nothing more than 1 damage, excluding modifiers. (I'll be excluding those for the rest of the answer) The tables in the player's handbook doesn't list a "1d1", just 1 damage. "1" is not a dice roll, it's a number.
Of course, the designers do seem intent on making unarmed strikes do 1 damage. I doubt it's a typo, especially considering the basic rules have been updated (it's on v0.2). They even supplied a feat (tavern brawler in PHB) and a class (monk) which increases the unarmed strike damage from 1+mod to some actual dice (like a 1d4). If they never intended the damage of unarmed strikes to be so lame, why would they provide that feat and that class?
As a side note: a critical hit from any weapon causes two failed death saving throws... including unarmed strikes. This is possibly the only time they are on par with more traditional weapons.
Possible RAI and RAF; The Spirit of Critical Hits
Critical hits seem to be meant to do more damage than a normal hit. This is reflected in the doubling of an attacks damage dice. It makes you generally do more damage than normal, although there is a slim chance that you don't do as much on that strike. You can see the odds for some weapon dice values here.
If you're looking to extend this logic to critical hits, I suggest instantiating a house rule for upping it to a 1d4 or supply 2 damage on a critical hit. This could be a fun rule, and it may alleviate the disappointment your players feel while getting that critical hit while hitting something with their... fists? ("Unarmed Strike" is really vague!)
However, it does seem that the designers really did want unarmed strikes to not hit hard unless you've had some experience. Once again, this is because of the Tavern Brawler feat in PHB and the monk class. Is this because they want people to use weapons unless they're a monk or a tavern brawler? Were they afraid of things like the grapplemancer builds coming into 5e? Hard to say. You could ask wizards of the coast by emailing sageadvice@wizards.com.
A Note
5e greatly acknowledges the importance of house rules in D&D culture, so I would not stick with the RAW reading of this. In that article linked above, written by Jeremy Crawford, one of the two lead designers for 5e, says:
We expect DMs to depart from the rules when running a particular campaign or when seeking the greatest happiness for a certain group of players.
The RAW just is not fun and takes the joy out of a critical hit. Come up with a rule that seems right (doubling to 2 damage, or going to 1d4 damage, or something else) for your group. In the words of Jeremy Crawford:
Fifth edition now belongs to the thousands of groups playing it.
Modify 5e to have fun in your group!
Best Answer
Yes, they need to know if an attack is a critical hit
Although there are no rules regarding this as far as I know, as rolling in the open or behind a screen is optional, there are abilities in the game that are reliant on knowing if an attack was a critical hit or not.
As one example, the Grave Domain of the cleric class has the Sentinel at Death's Door feature (XGtE, p. 20):
If the player had no way to tell if an attack was a critical hit, this feature would be worthless.
From a DM point of view, I would argue that the players should know when an attack is a crit, for a number of reasons. It builds tension and excitement, and it lets them use the abilities that might save them (the Lucky feat or the Rune Knight fighter's Runic Shield feature (TCoE, p. 45-46)) or not waste the abilities that won't (the shield spell).
Even if rolling in the open, the DM should still state that the hit was critical, as there are monsters and abilities that cause critical hits on values lower than a 20 on the die.