What kind of printer do sticker publishers use to make stickers that are on a sheet but diecut?

Answer 1

Don't know if this is applicable, but Avery makes all sorts of label materials that are ink-jet printable. Die cut on peel-off backing. Mailing labels, every sort.

Answer 2

They print the stickers first, then run them through the cutter. The office supply stores sell them pre-cut in sevearl different shapes like mailing labels and "thought bubbles", and they tell you how to line up your text to make it work with the cuts.