Depending on which edition you're playing, it varies.
In all cases, you may hand someone else an Artifact and if you end your attunement to it voluntarily, or a day passes without you touching your Artifact, they may attune to it in your place. The Hearthstone's behavior is what varies.
The rules in 1st and 2nd Edition are the same, so I'll only be quoting 2E and 3E as they are the clearest.
Attuning an Artifact
The nature of Attunement is that if an Exalted is unable to make physical contact with an Artifact for 1 full day, their attunement ends.
2nd Edition
the attunement dissipates if the character does not make daily skin contact with the item.
Exalted 2E Core Rules, p.380
3rd Edition
Attunement [...] and persists until the Exalt drops her commitment or is separated from the artifact for a full day.
Exalted 3E Core Rules, p. 594
In this case, clearly...if you give the Artifact to someone else and either break your attunement to it or it's away from you for a day, they can attune to it and use it
Hearthstones
Use of a Hearthstone requires you to be attuned to the Manse that generated the Hearthstone
Special Effects are the inherent powers of a Hearthstone--those things listed as part of their artifact writeup. Some of these require attunement to the Manse that generated the Hearthstone.
Exalted 2E
When an Exalt is attuned to a manse, places that manse’s hearthstone into an item made with one of the five magical materials and then invests Essence in the item to attune it to himself, the hearthstone resonates as well. Such a resonant hearthstone has a beneficial effect on the Exalt who carries it.
Exalted 2E Core Rules, p. 382
So, in 2nd Edition, the only use for a Hearthstone you are not attuned to is as an Essence Battery that you can plug into an Artifact powered by Hearthstones (that's the one use that doesn't require attunement to the parent manse).
Exalted 3E is more varied--Hearthstones provide two basic things: Essence Regen (if they are linked to a Manse, which they may not be) and Special Effects.
So, to the Essence Regen...
If a hearthstone is linked to a manse, and the manse’s owner places the hearthstone in a specially-designed socket in an attuned artifact, then the character enjoys the benefits of standing within the manse wherever she goes
Exalted 3E Core Rules, p.162
However, 3E is different in that many Hearthstones just Do Things with no requirement to be attuned to the manse--in fact, some Hearthstones aren't associated with a Manse at all
Many hearthstones influence the region around them. Others channel the power of Creation into more targeted and specific miracles
Exalted 3E Core Rules, p.603
From there, on reading through the listed Artifacts in the Exalted 3E Core Rule Book, it talks about them having to be in an "Attuned Hearthstone Socket" which is a hearthstone socket on an Artifact you are attuned to--it has nothing to do with being attuned to a Manse.
So, in short, in Third Edition anyone can use a Hearthstone's special properties either because it is an ambient, always-on effect or because it only requires you to socket it into an Attuned Artifact.
Attuning to a Manse
If you want to overcome these issues, you'll need to take the person you're gifting the Artifact to over to the Manse that Hearthstone came from, and let them attune to it...a time-consuming process that alerts the owner of the Manse and can be blocked by them (only if they are present, if you're playing 3E).
2nd Edition:
This attunement is a dramatic action requiring a number of hours equal to the site’s rating. If the demesne already has beings attuned to it, they mystically feel the ritual regardless of their current location, but they do not know who is attempting to attune to the demesne unless they are physically present. Unless every owner gives consent, the attunement automatically fails.
p.133
3rd Edition:
It takes six hours of meditation and careful co-mingling of Essence to attune to and claim ownership of an unclaimed demesne. The owner of a demesne can reflexively sense all attempts to attune to the demesne while present, and may automatically block them, permit them as sub-attunements which grant attunement benefits without bestowing ownership, or may even allow ownership to be taken. [...] While the owner remains attuned but not present, she may remotely sense attempts to attune to the demesne, but may not block them; they automatically succeed as sub-attunements.
p. 161
TL;DR:
They can attune to the Artifact with no problems.
If you're playing 2E, then the person you gifted the artifact to cannot make use of the Hearthstone without a trip to attune to the originating Manse.
If you're playing 3E, then the person you gifted the artifact to can use the Hearthstone's special powers, but (if it is Linked to a Manse) do not gain the Essence Regen unless they make a trip to attune to the originating Manse.