Oh, good question!
The wording does specify that the weapon is doing the casting, and from a reading of the weapon quality on its own merits that could be an argument that the weapon would damage the target and gain the temporary hit points (it doesn't actually 'heal' anything). It says the wielder permits it to cast the spell if they desire, it doesn't say they cast it or cast from it.
I will also note that the wording of a ring of spell storing specifically uses the wording 'that the wearer can cast' in regards to how the spells are cast, rather than 'the ring casts'. So wording-wise, it looks like the weapon is the caster, unlike wands and such, which say a spell is cast 'from the wand' rather than 'by the wand'.
Personally, I feel it's unintended wording, and as such, stored spells like charm person would charm the target towards the wielder, not just make them regard the weapon in a friendly light, but it's definitely something we'll need to test out and do some crowd-sourcing for more data on.
*loads up his spellstoring fork with vampiric touch and goes looking for blahpers*
"Hey! Guess what! Atalius has another question!"
