If someone has the spell armor of agathys on them, and get hit by an attack, to which they then cast shield to cause to miss them, does the retributive damage effect of armor of agathys trigger?

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Both of these spells trigger when you are hit. Therefore these are simultaneous effects. Lets look at the rule on simultaneous effects:

If two or more things happen at the same time on a character or monster's turn, the person at the game table — whether player or DM — who controls that creature decides the order in which those things happen.

This would suggest it depends on whoever's turn it is.

If it the enemy's turn and they get to decide the order, they'll of course determine you apply shield first which then prevents the hit entirely, therefore no retributive damage.

But if you were to provoke an attack on your own turn, you would get to determine the order and could decide the armor of agathys triggers first before the shield spell blocked it.

Opinions? Does it trigger damage every time? Never. or, depending on who's turn it is? What are yalls thoughts?

I'd say you'd have to resolve any reaction that can turn a hit into a miss before any reaction that triggers off of a hit. If you were to resolve Armor of Agathys first, then Shield, the latter could potentially turn a hit into a miss. This could result in a nonsensical scenario where you have an ability resolving off of being hit by an attack that didn't hit.

While the game often isn't logical from a narrative, in world point of view, it generally has to be logical from a mechanics, rules logic point of view. Being hit by an attack is a binary state; you either are hit, or are not hit. Shield is a reaction that occurs on a hit, but can turn it into a miss. Armor of Agathys only occurs on a hit; therefore it's logical you need to determine if the hit actually is a hit before triggering anything off of the attack hitting.

However, there's is nothing in RAW that says effects have to be resolved logically and in theory, using the quoted rule about timings, you could have a scenario where you have an ability triggering off of a hit that didn't actually hit.

So I'd say assumption of RAI being Shield first, then Armor of Agathys, but RAW order dictated by current persons turn.

Shield has 2 effects, firstly it increases tyour AC by 5, secondly you take no damage from magic missile.

First the easy one, if you had cast shiled before this attack (say increasing your AC from 18 to 23) and another melee attack against you comes in at 21 to hit, the attack misses and AoA has no effect.

If the attack is the one that initiates the casting of shield then it either hits the caster of shield when they have an AC of 18 (causing a hit and damage) or and AC of 23 (causing a miss). Clearly the first isn't intended (and if the person whose turn it is can cause the order they would choose that shield takes place after the attack) so it only makes sense for the attack to be a miss.

For magic missile you "take no damage" but it still hits, however as it is not a melee atack AoA has no effect.

I don't think an action and a reaction are considered to happen at the same time, consider the BBEG casting fireball and one of the party casting counterspell as a reaction, and the DM deciding that the fireball happens first so counterspell has no effect (other than using up a spell slot). A reaction sees something about to happen and acts before that event takes place.  I think the rule about things happening at the same time is mainly for things like when to characters are both reacting to the same trigger.

Quote from Jegpeg >>

I think the rule about things happening at the same time is mainly for things like when two characters are both reacting to the same trigger.

 Mhm. Same trigger.

"If a creature hits you with a melee attack" and "when you are hit by an attack".

I would lean on the Adjudicating Reaction Timing section of the DMG, which tells us that the Shield spell interrupts its trigger, whereas Armor of Agathys is applied after the hit.

Quote from Gruntler >>

I would lean on the Adjudicating Reaction Timing section of the DMG, which tells us that the Shield spell interrupts its trigger, whereas Armor of Agathys is applied after the hit.

 Oh wow, I completely forgot about this rule, good spot! DMGnomes are at it again, sneaking rules into my books!

Quote from Gruntler >>

I would lean on the Adjudicating Reaction Timing section of the DMG, which tells us that the Shield spell interrupts its trigger, whereas Armor of Agathys is applied after the hit.

Beautiful answer, thanks! This really does zero in on the difference between the two triggers despite their seemingly same-ness, one is from a reaction and that has special timing associated with it.