The electronic and geometrical structures of the water dimer anion are investigated by density functional calculations. Theoretical values of the very weak electron affinities and vertical detachment energies show good agreement with available experimental data. It is predicted that a significant modification of the neutral parent geometry is introduced by the electron attachment process and leads to a large excitation of anion intermolecular modes. The interpretation of field-detachment experiments of (H2O)(2)(-) and (D2O)(2)(-), produced in collisions between laser-excited Rydberg atoms and cold neutral water clusters, confirms the existence of large amplitude low-frequency motions in the anions.