We perform magnetohydrodynamic simulations of accreting, equal-mass binary black holes in full general relativity focusing on the impact of black hole spin on the dynamical formation and evolution of minidisks. We find that during the late inspiral the sizes of minidisks are primarily determined by the interplay between the tidal field and the effective innermost stable orbit around each black hole. Our calculations support that a minidisk forms when the Hill sphere around each black hole is significantly larger than the black hole's effective innermost stable orbit. As the binary inspirals, the radius of the Hill sphere decreases, and minidisks consequently shrink in size. As a result, electromagnetic signatures associated with minidisks may be expected to gradually disappear prior to merger when there are no more stable orbits within the Hill sphere. In particular, a gradual disappearance of a hard electromagnetic component in the spectrum of such systems could provide a characteristic signature of merging black hole binaries. For a binary of given total mass, the timescale to minidisk 'evaporation' should therefore depend on the black hole spins and the mass ratio. We also demonstrate that accreting binary black holes with spin have a higher efficiency for converting accretion power to jet luminosity. These results could provide new ways to estimate black hole spins in the future.