Source is everything! Should leave it at that, but in the interests of sticking my neck out:
If the possession is granting the bonus directly (e.g. "Remove
to make bearer strength +1"), that bonus is ignored. The source is the possession.
Sometimes the source is another card (e.g. "Bearer gains
hunter 1"). Mighty Longsword basically makes its bearer gain the text "While skirmishing a non-hunter character, this character is strength +1." So the minion is actually the source of the strength bonus, and the bonus stays.