Deinking paper begins with pulping, a process that closely resembles virgin fiber pulping. Fibers swell in the water, and as they rub against each other the impurities start to fall away. Heat and chemicals speed the process. Screening then removes large, heavy contaminants, and fine screening removes the smaller, lighter ones. Next the pulp is washed, similar to the way clothes are washed.
Flotation is another deinking method.
It's the opposite of washing, in that the ink is turned back to its water-repelling state. Foaming agents create air bubbles that attract the ink particles. The air bubbles and ink rise to the surface, and are floated off. Any inks left after washing and flotation may be dispersed or bleached. Dispersion breaks the ink particles into tiny bits that stay in the pulp but that are invisible to the eye.