Neither. Armor Training only applies to worn armor; shields are not worn armor.
To correct your premise: For the purposes of Armor Training, a shield is not considered a piece of armor. Armor and shields both give AC (armor class) bonuses, but they are entirely separate! Armor is worn in your "armor" slot, and a shield is a thing you hold. That is why a Shield bonus to AC will stack with an Armor bonus to AC. For example, if a character wears chainmail and wields two shields, then "each piece of armor" only includes the chainmail.
In the Fighter class description in the CRB, the text on Armor Training says the following:
Whenever he is wearing armor, he reduces the armor check penalty by 1 (to a minimum of 0) and increases the maximum Dexterity bonus allowed by his armor by 1. Every four levels thereafter (7th, 11th, and 15th), these bonuses increase by +1 each time, to a maximum –4 reduction of the armor check penalty and a +4 increase of the maximum Dexterity bonus allowed.
Unfortunately, the effects of Armor Training don't apply to both the armor and shield, because Armor Training does not apply to shields. Armor Training specifically adjusts the ACP and max Dex of worn armor. This class feature is more useful to Fighters clad in medium or heavy armor.
Example: Felix the Fighter wears a +1 Mithral Breastplate (0 ACP, +6 max Dex bonus). He also carries a standard tower shield (-10 ACP, +2 max Dex bonus). His Armor Training increases the breastplate's max Dex bonus, but it can't reduce the breastplate's ACP below 0. The shield is unaffected by Armor Training, and causes him to take an ACP penalty of -10, and also reduces his maximum Dex bonus to +2.
Also, if you're using the alternate rules for piecemeal armor from Ultimate Combat, then Armor Training will affect the worst ACP and lowest max Dex bonus that is determined by armor:
If a character is wearing more than one armor piece, she add the armor costs, armor bonuses, and weights of the armor pieces, and takes the worst maximum Dexterity bonus, arcane spell failure chance, and speed limitations from among the various armor pieces to determine the full statistics and qualities of the armor she is wearing.
When/if Felix is wearing multiple pieces or armor (or multiple armors, for some reason), this is how his Armor Training applies: He takes the most negative ACP of any piece he's wearing, and takes the lowest max Dex allowed by any piece he's wearing - these may be different pieces. Then, the benefits from his Armor Training apply to those ACP and max Dex bonus.