Thaumaturgy uses the body's own aether to cast spells - as can be seen in the class questline with the sibling of the guildmasters who want to learn thaumaturgy, but has too low aether.
Conjury borrows aether from the surroundings - likely in addition to using the caster's own aether since that can also be used, but apparently does so at a low enough level that it is not considered harmful to the surroundings.

White magic and Black magic draws aether from the surroundings of the caster, and when used on a large enough scale can be highly damaging to the world - just like primals are a danger because they drain aether from the world around them. Used on a small scale they are no problem - the world will quickly replace the aether used.

Red mages use their own aether, but as long as they don't use too much at once it won't be harmful to them. They will recover their own aether in between as long as they don't go overboard with the usage.

The crafting classes all use external aether in the form of crystals to help their crafting. (Fire crystals for heating, wind for drying, etc. etc.)

For the rest of the DoW/DoM classes/jobs it is never explicitly explained just how they use aether - we only assume they do since some of their abilities would otherwise be physically impossible.
The DoW classes almost certainly use their own aether. The DoM classes probably do so as well.