Consider that I am possessing a host using magic jar, and that while I am possessing the host, I am subjected to the feeblemind spell and fail my saving throw.
For whatever reason after this, I decide to the leave the host body and return to the container of magic jar. Well, now a question arises: who continues to suffer the effects of feeblemind?
When you possess a body using Magic Jar the following applies (taken from the spell description):
Magic Jar (SRD):
Your body falls into a catatonic state as your soul leaves it and enters the container you used for the spell’s material component.
... your soul moves into the target’s body, and the target’s soul becomes trapped in the container.
Once you possess a creature’s body, you control it. Your game statistics are replaced by the statistics of the creature, though you retain your alignment and your Intelligence, Wisdom, and Charisma scores.
While possessing a body, you can use your action to return from the host body to the container if it is within 100 feet of you, returning the host creature’s soul to its body.
When that possessed creature is targeted by a Feeblemind spell the following applies (taken from the spell description):
Feeblemind (SRD)
You blast the mind of a creature that you can see within range, attempting to shatter its intellect and personality. The target takes 4d6 psychic damage and must make an Intelligence saving throw.
On a failed save, the creature’s Intelligence and Charisma scores become 1. The creature can’t cast spells, activate magic items, understand language, or communicate in any intelligible way. The creature can, however, identify its friends, follow them, and even protect them.
So you gain the hit points of the host, but you retain your alignment and your Intelligence, Wisdom, and Charisma scores.
The hit point damage done by a Feeblemind are therefore done to the host's hit points. However the damage done by the Feeblemind to the Int, Wis, and Cha scores of the target creature are yours, by definition of the Magic Jar spell.
When you return to the container used in the Magic Jar spell there is nothing saying that you recover in any way, so by the RAW you are left in the container with Int, Wis, and Cha of 1.
This also fits in a "common sense" way. For it to happen in any other way would imply that it is not actually the caster of the Magic Jar that is possessing the target, that it is some kind of mind control, or mind copy, or something else which just does not fit the spell descriptions, the "feel" of the spells, or the thematic context in any way, in my opinion.
Another thing to consider is that the Magic Jar spell says your soul moves into the host body. The soul is generally regarded as the essence of a person, so this is not mere control of a body from a distance, the caster's very essence moves out of their body and into the host's to possess target creatures.
The text of the Soul Jar spell says that:
... your soul moves into the target’s body, and the target’s soul becomes trapped in the container.
The nearest comparison I can find is that of the lich, that stores its soul in a phylactery.
From the The Monster Manual (p203):
A lich is created by an arcane ritual that traps the wizard's soul within a phylactery. Doing so binds the soul to the mortal world, preventing it from traveling to the Outer Planes after death. A phylactery is traditionally an amulet in the shape of a small box, but it can take the form of any item possessing an interior space into which arcane sigils of naming, binding, immortality, and dark magic are scribed in silver.
With its phylactery prepared, the future lich drinks a potion of transformation-a vile concoction of poison mixed with the blood of a sentient creature whose soul is sacrificed to the phylactery. The wizard falls dead, then rises as a lich as its soul is drawn into the phylactery, where it forever remains.
Arguably, although the caster of Soul Jar is not a lich, the example of the lich seems like it sets a precedent. The difference being that a lich wholly and permanently divorces their mind and body from the soul, whereas the caster of Soul Jar puts their soul into another being.
If the soul were neccessary for targetting creatures with spells that affect the mind, you would have to cast it on the phylactery not the lich's body where the mind is housed. There are no additional rules provided for the Lich that distinguish between it's mind and body.
By analogy, the caster of Soul Jar does not bring their mind with them when they possess the body, therefore they are not neccessarily subject to spells that effect the mind.
That being said, this line of reasoning can open up dangerous precedents that might be better for a GM to judge. What constitutes a spell that affects the mind? Does this make the possessor immune to mental magic? Clearly this requires a GM as the final arbiter.
