  The electromagnetic form factors $G_{\rm E}$ and $G_{\rm M}$ of the proton
and neutron in the timelike region are extracted in a study of the processes
$e^+ e^-\to \bar{p}p$ and $e^+ e^-\to \bar{n}n$. The reaction amplitude is
evaluated within the distorted wave Born approximation, with the interaction of
the antinucleon-nucleon ($\bar{N}N$) pair taken into account. The latter is
constructed within $SU(3)$ chiral effective field theory up to the
next-to-leading order. An excellent description of the $e^+ e^-\to \bar{N}N$
data in the energy region from the $\bar{N}N$ threshold up to center-of-mass
energies $E_{\rm cm}=2.2$~GeV is achieved. Results for the electromagnetic form
factors $G_{\rm E}, G_{\rm M}$, $G_{\rm E}/G_{\rm M}$, and the subtracted
effective form factors, $G_{\rm osc}$, are provided. These can be helpful for
further studies of the properties of the nucleons.
