  We investigate the possibility of disentangling different new physics
contributions to the rare meson decays $K\rightarrow\pi+\displaystyle{\not}E$
and $B\rightarrow K(K^*)+\displaystyle{\not}E$ through kinematic distributions
in the missing energy $\displaystyle{\not}E$. We employ dimension-$6$ operators
within the Low-Energy Effective Field Theory (LEFT), identifying the invisible
part of the final state as either active or sterile neutrinos. Special emphasis
is given to lepton-number violating (LNV) operators with scalar and tensor
currents. We show analytically that contributions from scalar, vector, and
tensor quark currents can be uniquely determined from experimental data of
kinematic distributions. In addition, we present new correlations of branching
ratios for $K$ and $B$-decays involving scalar and tensor currents. As there
could a priori also be new invisible particles in the final states, we include
dark-sector operators giving rise to two dark scalars, fermions, or vectors in
the final state. In this context, we present new calculations of the inclusive
decay rate $B\rightarrow X_s+\displaystyle{\not}E$ for dark operators. We show
that careful measurements of kinematic distributions make it theoretically
possible to disentangle the contribution from LEFT operators from most of the
dark-sector operators, even when multiple operators are contributing. We
revisit sum rules for vector currents in LEFT and show that the latter are also
satisfied in some new dark-physics scenarios that could mimic LEFT. Finally, we
point out that an excess in rare meson decays consistent with a LNV hypothesis
would point towards highly flavor non-democratic physics in the UV, and could
put high-scale leptogenesis under tension.
