They are killed and skinned. The problem with shearing live weasels would be that the hairs' natural taper is part of what makes a good brush (especially the wee fine point types we like). Shave your weasel once, and the next, "crop" will be mostly follicles with blunt ends.
Oh they are killed, but winsor and newton alone ,let alone the wider art industry just couldn't afford to buy skins from trapped wild ones.there are boutique high end brush makers who do still buy their furs from fur trappers, hand selecting them and such, but not a relatively large company such as W&N.
(btw, if you go to their site you can request free samples. i got free oil bars from them which i have since used up and restocked because of how good they were.when it asks what medium you work in ,choose acrylic if you want paint, or oil on the off-change they send you a good medium for oil washes.)