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Exchange interactions between isolated pairs of spin centers in diamond have been calculated, based on an accurate atomistic electronic structure for diamond and any impurity atoms, for spin-center separations up to 2~nm. The exchange interactions exceed dipolar interactions for spin center separations less than 3~nm. NV$^-$ spin centers, which are extended defects, interact very differently depending on the relative orientations of the symmetry axis of the spin center and the radius vector connecting the pair. Exchange interactions between transition-metal dopants behave similarly to those of NV$^-$ centers. The Mn\---Mn exchange interaction decays with a much longer length scale than the Cr\---Cr and Ni\---Ni exchange interactions, exceeding dipolar interactions for Mn\---Mn separations less than 5~nm. Calculations of these highly anisotropic and spin-center-dependent interactions provide the potential for design of the spin-spin interactions for novel nanomagnetic structures., 5 pages