  Near-threshold $\Lambda\bar\Lambda$ mass spectra for the reactions $e^+e^-
\to \eta\Lambda\bar\Lambda$ and $e^+e^- \to \phi\Lambda\bar\Lambda$ are
investigated with an emphasis on the role played by the interaction in the
$\Lambda\bar\Lambda$ system. A variety of $\Lambda\bar\Lambda$ potential models
is employed that have been established in the analysis of data on $p\bar p\to
\Lambda\bar\Lambda$ in the past. It is shown that the near-threshold
enhancement observed for the two $e^+e^-$ reactions can be reproduced by
considering the $\Lambda\bar\Lambda$ final-state interaction in the partial
waves suggested by the helicity-angle analysis of the experiments. For $e^+e^-
\to \eta\Lambda\bar\Lambda$ the same $\Lambda\bar\Lambda$ $S$-wave interaction
as in $e^+e^- \to \Lambda\bar\Lambda$ is relevant and with it a consistent
description of the pertinent measurements can be achieved. It is pointed out
that a nonzero threshold cross section as observed for the latter reaction is
not supported by the new $\eta\Lambda\bar\Lambda$ data.
