Taken from the description of concordance on DMG page 164:
An artifact’s concordance score measures the artifact’s attitude toward its wielder. The scale ranges from 0 (angered) to 20 (pleased).
When a character takes possession of an artifact, it starts with a concordance of 5. (The owner’s race, class, or other characteristics might adjust this starting concordance.) Various actions and events increase or decrease this score as long as the character possesses the artifact. When the artifact is pleased with its wielder’s actions, its concordance goes up. When the wielder acts contrary to the artifact’s desires, its concordance decreases.
Since concordance is the artifact's attitude toward its wielder, an artifact should be considered as having a different concordance score for each individual wielder.
Suppose Alphonse acquires a Deck of Many Things, and raises its concordance with him to 12 - the Deck of Many Things is pleased with him! If Alphonse then hands the Deck of Many Things to Bertha, Bertha's now its wielder. The Deck of Many Things has no opinion of her yet, so its concordance with her is 5. When Bertha hands the Deck of Many Things back to Alphonse in the future, its concordance with him is still 12 like it used to be.
A special case: shared artifacts
Some artifacts are designed to be shared between all party members. In this case, the DM should consider treating the artifact's "wielder" as the entire party for most intents.
An example: Finding new owners
Many artifacts explicitly try to move on to new owners, such as the Orb of Light, whose Moving On description says this:
"We have made the world a better place." One morning, the Orb is gone, replaced by a luminescent (but mundane) pearl worth 5,000 gp. The former owner gains a permanent +1 bonus to AC against attacks by undead creatures.If the owner failed the Orb, the next time the character lifts it, the artifact crumbles into coal, leaving permanent marks on the character's hand. The former owner takes a permanent -1 penalty to AC against attacks by undead creatures. The Orb reappears elsewhere in the world, ready to be claimed by another aspirant.
Note that last sentence: the Orb decides it should seek out a new owner eventually. Every new artifact you find, including the Orb of Light, starts at a concordance around 5, despite the fact that just yesterday, it might have been possessed by someone who had a concordance of 50+ with it. If its concordance was absolute between all wielders, an artifact's starting score should vary heavily depending on who its previous wielder was - but it doesn't vary. It starts at around 5 regardless of who it was previously owned by.
New owner, new concordance.