Graphene's near-field radiative heat transfer is determined from its electrical conductivity, which is commonly modeled using the local (wave-vector independent) Kubo and Drude formulas. In this paper, we analyze the nonlocality of the graphene electrical conductivity using the Lindhard model combined with the Mermin relaxation time approximation. We also study how the variation of the electrical conductivity with the wave vector affects near-field radiative conductance between two graphene sheets separated by a vacuum gap. It is shown that the variation of the electrical conductivity with the wave vector, , is appreciable for greater than , where is the magnitude of the wave vector in the free space. The Kubo model is obtained by assuming , and thus is not valid for . The Kubo electrical conductivity provides an accurate estimation of the spectral radiative conductance between two graphene sheets except for around the surface-plasmon-polariton frequency of graphene and at separation gaps smaller than 20 nm where there is a non-negligible contribution from electromagnetic modes with to the radiative conductance. The Drude formula proves to be inaccurate for modeling the electrical conductivity and radiative conductance of graphene except for at temperatures much below the Fermi temperature and frequencies much smaller than , where and are the chemical potential and reduced Planck's constant, respectively. It is also shown that the electronic scattering processes should be considered in the Lindhard model properly, such that the local electron number is conserved. A simple substitution of by (, and being the angular frequency, imaginary unit, and scattering rate, respectively) in the collisionless Lindhard model does not satisfy the conservation of the local electron number and results in significant errors in computing the electrical conductivity and radiative conductance of graphene.