Multiply ionized fullerene dimers ([C(60)](2) (q+),q=1-6) have been studied by means of state-of-the-art density functional theory methods. We found that the singly charged dimer is more strongly bound than the corresponding neutral van der Waals dimer at the binding distance of the latter; in contrast, multiply charged dimers (q>or=2) are unstable. For the latter dimers, the calculated kinetic energy release in the binary fission yielding intact fullerenes is lower than those reported in recent experimental work. This implies that, in such experiments, there are significant internal excitations in the separating monomers. We also show that electron transfer within the charged dimers occurs on the subfemtosecond time scale, in accordance with the high charge mobility observed in dimers and larger clusters of fullerenes. This provides an explanation for the even-odd effects in the measured multiple ionization yields of fullerene dimers.