It follows the rules of use magic item, so the person with the item's concentration. As the SRD (and DMG p. 141) states:

Some magic items allow the user to cast a spell from the item. ... the user of the item must concentrate if the spell requires concentration.

This follows with Mike Mearls Twitter post (while Mike isn't offical RAW, we are talking about UA material he had a hand in writing, and it does follow rules for other activated magic items):

whoever activates it must concentrate

https://twitter.com/mikemearls/status/818582158994325504?ref_src=twsrc%5Etfw

Various topics between the two versions vary.

Guidance and Prerequisites

Unearthed Arcana

For a player character to gain a sidekick, there are some prerequisites to be met. The sidekick can be any creature from the Monster Manual or other D&D book, and these prerequisites are necessary:

• Its challenge rating must be 1 or lower.
• The two of you must be friends.

There is no mention of minimal or maximum party-size (aka number of players) that limits the number of sidekicks in a group of PCs.

Both the Expert and Spellcaster classes requires the creature to "have at least one language in its stat block that it can speak".

Essentials Kit

In the adventure booklet included in the Essentials Kit – Dragon of Icespire Peak – the Running the Adventure chapter recommends having a sidekick if you, as the DM, are running the adventure for a single player. The sidekick must be one chosen from the Sidekick Cards included in the box.

The characters presented in the Sidekick Cards are humanoids of various races with at least one level of one of the Sidekick classes. There are no other requirements in this version.

Leveling Up

Unearthed Arcana

A sidekick gains a level whenever the associated PC gains a level.

Whenever you gain a level, your sidekick also gains a level. It doesn’t matter how much of your recent adventures the sidekick experienced; the sidekick levels up because of a combination of the adventures it shares with you and its own training.

Essentials Kit

The Sidekick explicitly counts as a PC for the purpose of sharing experience points between the party members. The Sidekick rules says:

As you and your sidekick adventure together, your sidekick gains experience points and reaches new levels the same way a player character does [...]

Hit Points

Unearthed Arcana

Whenever a Sidekick gains a level, it gains a Hit Die based on its creature size (d4 for Tiny, d6 for Small, d8 for Medium, etc).

Essentials Kit

The creature Hit Die is always a d8, as the Appendix deals only with the Sidekicks presented in the Sidekick Cards in the box, which are all humanoids.

Ability Score Improvement

Unearthed Arcana

There is no guidance regarding which ability score you should increase when a sidekick gains such feature.

Essentials Kit

For each of the three classes, there is guidance to what ability score should be increased. For Warrior is +2 to Strength, for Expert is +2 to Dexterity, for Spellcaster is +2 to Intelligence or Wisdom, depending on its spellcasting ability.

Sidekick Classes

The classes available to sidekicks are the same for both UA and Essentials: Warrior, Expert, and Spellcaster. They, however, have different features for each level.

Unearthed Arcana

The UA classes have a complete table of features, from levels 1 to 20.

Essentials Kit

The Essentials Kit only presents a table of features per level ranging from levels 1 to 6. This limitation is due to the adventure included in the box: Dragon of Icespire Peak is meant to be played from levels 1 to 6.

Warrior

This section compares the levels 1-6 of the Warrior class.

\begin{array}{|c|c|l|l|} \hline \textbf{Level} & \textbf{Prof. Bonus} & \textbf{Unearthed Arcana} & \textbf{Essentials Kit} \\ \hline \text{1st} & \text{+2} & \text{Second Wind, Bonus Proficiencies} & \text{Martial Role, Selected Proficiencies*} \\ \hline \text{2nd} & \text{+2} & \text{Danger Sense} & \text{Second Wind} \\ \hline \text{3rd} & \text{+2} & \text{Improved Critical} & \text{Improved Critical} \\ \hline \text{4th} & \text{+2} & \text{Ability Score Improvement} & \text{Ability Score Improvement} \\ \hline \text{5th} & \text{+3} & \text{Extra Attack} & \text{–} \\ \hline \text{6th} & \text{+3} & \text{Ability Score Improvement} & \text{Extra Attack} \\ \hline \end{array}

*By "Selected Proficiencies", I mean a preselected set of proficiencies. The Warrior from the Essentials Kit has the following proficiencies already chosen:

• Saving Throws: Constitution
• Skills: Athletics, Perception, Survival

Both versions are proficient with all armor, shields, and simple and martial weapons.

Expert

This section compares the levels 1-6 of the Expert class.

\begin{array}{|c|c|l|l|} \hline \textbf{Level} & \textbf{Prof. Bonus} & \textbf{Unearthed Arcana} & \textbf{Essentials Kit} \\ \hline \text{1st} & \text{+2} & \text{Expertise, Helpful, Bonus Proficiencies} & \text{Helpful, Tools, Selected Proficiencies*} \\ \hline \text{2nd} & \text{+2} & \text{Cunning Action} & \text{Cunning Action} \\ \hline \text{3rd} & \text{+2} & \text{Jack of Many Trades} & \text{Expertise} \\ \hline \text{4th} & \text{+2} & \text{Ability Score Improvement} & \text{Ability Score Improvement} \\ \hline \text{5th} & \text{+3} & \text{Extra Attack} & \text{–} \\ \hline \text{6th} & \text{+3} & \text{Expertise} & \text{Extra Attack} \\ \hline \end{array}

*By "Selected Proficiencies", I mean a preselected set of proficiencies. The Expert from the Essentials Kit has the following proficiencies already chosen:

• Saving Throws: Dexterity
• Skills: Acrobatics, Performance, Persuasion, Sleight of Hand, Stealth

Both versions have proficiency with light armor and all simple weapons. The Unearthed Arcana version also gains proficiency with two tools of your choice, while the Essentials Kit version gains proficiency with rapiers, shortswords, and possibly with thieves' tools and a musical instrument (these last two from the "Tools" trait in its stat block).

Spellcaster

This section compares the levels 1-6 of the Spellcaster class.

\begin{array}{|c|c|l|l|} \hline \textbf{Level} & \textbf{Prof. Bonus} & \textbf{Unearthed Arcana} & \textbf{Essentials Kit} \\ \hline \text{1st} & \text{+2} & \text{Spellcasting, Bonus Proficiencies} & \text{Magical Role*, Spellcasting**, Selected Proficiencies***} \\ \hline \text{2nd} & \text{+2} & \text{Magical Recovery} & \text{–} \\ \hline \text{3rd} & \text{+2} & \text{–} & \text{–} \\ \hline \text{4th} & \text{+2} & \text{Ability Score Improvement} & \text{Ability Score Improvement} \\ \hline \text{5th} & \text{+3} & \text{–} & \text{–} \\ \hline \text{6th} & \text{+3} & \text{Potent Cantrips} & \text{Potent Cantrips} \\ \hline \end{array}

*The Essentials Kit Spellcaster has the Magical Role feature, which defines the Spellcasting Ability for the sidekick character. It only allows for Intelligence or Wisdom to be chosen.

In the Unearthed Arcana version, the Spellcasting Ability is chosen in the Spellcasting feature, with Charisma being a possible Spellcasting Ability too.

**Each version of the Spellcasting feature has a different spell slot progression (shown below), along with other differences:

• The Unearthed Arcana version has a spell list associated to one of the PCs classes that can conjure up to 9th level spells (Warlock included), and a Spellcasting Focus associated with such class. Spells and cantrips can be chosen freely from the associated spell list, following the progression below (the last three columns represent the Spell Slots):

\begin{array}{|c|c|c|c|c|c|} \hline \textbf{Level} & \textbf{Cantrips Known} & \textbf{Spells Known} & \textbf{1st} & \textbf{2nd} & \textbf{3rd} \\ \hline \text{1st} & \text{3} & \text{2} & \text{2} & \text{–} & \text{–} \\ \hline \text{2nd} & \text{3} & \text{3} & \text{3} & \text{–} & \text{–} \\ \hline \text{3rd} & \text{3} & \text{4} & \text{4} & \text{2} & \text{–} \\ \hline \text{4th} & \text{4} & \text{5} & \text{4} & \text{3} & \text{–} \\ \hline \text{5th} & \text{4} & \text{6} & \text{4} & \text{3} & \text{2} \\ \hline \text{6th} & \text{4} & \text{7} & \text{4} & \text{3} & \text{3} \\ \hline \end{array}

• The Essentials Kit version has no spell list associated, independent of the chosen Magical Role. Both spells and cantrips are previously chosen for each role, and it has no Spellcasting Focus. It follows the progression below (the last two columns represent the Spell Slots):

\begin{array}{|c|c|c|c|c|} \hline \textbf{Level} & \textbf{Cantrips Known} & \textbf{Spells Known} & \textbf{1st} & \textbf{2nd} \\ \hline \text{1st} & \text{2} & \text{1} & \text{2} & \text{–} \\ \hline \text{2nd} & \text{2} & \text{2} & \text{2} & \text{–} \\ \hline \text{3rd} & \text{2} & \text{3} & \text{3} & \text{–} \\ \hline \text{4th} & \text{3} & \text{3} & \text{3} & \text{–} \\ \hline \text{5th} & \text{3} & \text{4} & \text{4} & \text{2} \\ \hline \text{6th} & \text{3} & \text{4} & \text{4} & \text{2} \\ \hline \end{array}

***By "Selected Proficiencies", I mean a preselected set of proficiencies. The Spellcaster from the Essentials Kit has the following proficiencies already chosen:

• Saving Throws: Wisdom
• Skills: Arcana, Investigation, Religion

Both versions are proficient with light armor and simple weapons.

Other details

Finally, it is important to note that the Unearthed Arcana content is playtest material, and therefore, can be unbalanced. The Essentials Kit version is official, but the lack of progress after level 6 limits the official rules for Sidekicks.