We study the baryonic charmonium decays of B mesons B+-->etacK+ and B+-->J/psiK+, where the etac and J/psi subsequently decay into a pp[over] or LambdaLambda[over] pair. We measure the J/psi-->pp[over] and LambdaLambda[over] anisotropy parameters alphaB=-0.60+/-0.13+/-0.14 (pp[over]), -0.44+/-0.51+/-0.31 (LambdaLambda[over ]) and compare to results from e;{+}e;{-}-->J/psi formation experiments. We also report the first observation of etac-->LambdaLambda[over]. The measured branching fraction is B(etac-->LambdaLambda[over ])=(0.87(+0.24)/(-0.21)(stat)(+0.09/-0.14) (syst)+/-0.27(PDG))x10-3. This study is based on a 357 fb-1 data sample recorded on the Upsilon(4S) resonance with the Belle detector at the KEKB asymmetric-energy e+ e- collider.