Background: Neutrino-pair bremsstrahlung processes from nucleon-nucleon scattering $NN\ensuremath{\nu}\overline{\ensuremath{\nu}}$ ($nn\ensuremath{\nu}\overline{\ensuremath{\nu}},\phantom{\rule{0.16em}{0ex}}pp\ensuremath{\nu}\overline{\ensuremath{\nu}}$, and $np\ensuremath{\nu}\overline{\ensuremath{\nu}}$) have recently attracted attention in studies of neutrino emission in neutron stars, because of the implications for the neutron star cooling. The calculated $NN\ensuremath{\nu}\overline{\ensuremath{\nu}}$ emissivities within the neutron star environment are relatively insensitive to the two-nucleon dynamical model used in the calculations, but differ significantly from those obtained using an one-pion-exchange (OPE) model.Purpose: We investigate the free $NN\ensuremath{\nu}\overline{\ensuremath{\nu}}$ cross sections using a realistic nucleon-nucleon scattering amplitude, comparing the relative sizes of the cross sections for the three processes $nn\ensuremath{\nu}\overline{\ensuremath{\nu}},\phantom{\rule{0.16em}{0ex}}pp\ensuremath{\nu}\overline{\ensuremath{\nu}}$, and $np\ensuremath{\nu}\overline{\ensuremath{\nu}}$.Method: We employ a realistic one-boson-exchange (ROBE) model for $NN$ scattering and combine those strong scattering amplitudes with the well-known nucleon weak interaction vertices to construct weak bremsstrahlung amplitudes. Using the resulting $NN\ensuremath{\nu}\overline{\ensuremath{\nu}}$ amplitudes we investigate the relative importance of the vector $({\mathrm{\ensuremath{\Gamma}}}_{V}^{\ensuremath{\mu}})$, axial vector $({\mathrm{\ensuremath{\Gamma}}}_{A}^{\ensuremath{\mu}})$, and tensor $({\mathrm{\ensuremath{\Gamma}}}_{T}^{\ensuremath{\mu}})$ terms. The ROBE model bremsstrahlung amplitudes are also used as a two-nucleon dynamical model with which we calculate the cross sections $\frac{d\ensuremath{\sigma}}{d\ensuremath{\omega}}$ for $nn\ensuremath{\nu}\overline{\ensuremath{\nu}},\phantom{\rule{0.16em}{0ex}}pp\ensuremath{\nu}\overline{\ensuremath{\nu}}$, and $np\ensuremath{\nu}\overline{\ensuremath{\nu}}$.Results: The three free $NN\ensuremath{\nu}\overline{\ensuremath{\nu}}$ cross sections $\frac{d\ensuremath{\sigma}}{d\ensuremath{\omega}}$ are of similar order of magnitude. Each increases with increasing neutrino-pair energy $\ensuremath{\omega}$. For the neutrino-pair energy of $\ensuremath{\omega}=1\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$ our $nn\ensuremath{\nu}\overline{\ensuremath{\nu}}$ results are in quantitative agreement with those previously reported by Timmermans et al. [Phys. Rev. C 65, 064007 (2002)], who used the leading-order term of the soft-neutrino-pair bremsstrahlung amplitude to calculate the cross sections. Differences between the $nn\ensuremath{\nu}\overline{\ensuremath{\nu}}$ and $pp\ensuremath{\nu}\overline{\ensuremath{\nu}}$ cross section are not discernible over the nucleon-nucleon incident energy region considered, due to the complete dominance of the axial vector component of the weak interaction nucleon vertex function ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{\mu}}$ as demonstrated analytically in Appendix A. The $np\ensuremath{\nu}\overline{\ensuremath{\nu}}$ cross section is smaller than either the $nn\ensuremath{\nu}\overline{\ensuremath{\nu}}$ or the $pp\ensuremath{\nu}\overline{\ensuremath{\nu}}$ cross section for low to moderate values of $\ensuremath{\omega}$; this characteristic only changes at larger neutrino-pair energies around $\ensuremath{\omega}\ensuremath{\sim}50\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$, which is above the low energy region characterized in Appendix B.Conclusions: The free $NN\ensuremath{\nu}\overline{\ensuremath{\nu}}$ cross sections, calculated using a realistic nucleon-nucleon amplitude model, are new except for the $nn\ensuremath{\nu}\overline{\ensuremath{\nu}}$ cross section at $\ensuremath{\omega}=1\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$ that was first reported by Timmermans et al., and at $\ensuremath{\omega}=0.5,1,2\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$ by Li et al. [Phys. Rev. C 80, 035505 (2009)]. The $nn\ensuremath{\nu}\overline{\ensuremath{\nu}}$ and $pp\ensuremath{\nu}\overline{\ensuremath{\nu}}$ cross sections are virtually identical in magnitude. All three $NN\ensuremath{\nu}\overline{\ensuremath{\nu}}$ processes are dominated by the axial vector component of the vertex function ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{\mu}}$, with only slight deviations from this behavior being seen in the $np\ensuremath{\nu}\overline{\ensuremath{\nu}}$ process at large neutrino-pair energies.