  We study the spatial decay of eigenfunctions of non-local Schr\"odinger
operators whose kinetic terms are generators of symmetric jump-paring L\'evy
processes with Kato-class potentials decaying at infinity. This class of
processes has the property that the intensity of single large jumps dominates
the intensity of all multiple large jumps. We find that the decay rates of
eigenfunctions depend on the process via specific preference rates in
particular jump scenarios, and depend on the potential through the distance of
the corresponding eigenvalue from the edge of the continuous spectrum. We prove
that the conditions of the jump-paring class imply that for all eigenvalues the
corresponding positive eigenfunctions decay at most as rapidly as the L\'evy
intensity. This condition is sharp in the sense that if the jump-paring
property fails to hold, then eigenfunction decay becomes slower than the decay
of the L\'evy intensity. We furthermore prove that under reasonable conditions
the L\'evy intensity also governs the upper bounds of eigenfunctions, and
ground states are comparable with it, i.e., two-sided bounds hold. As an
interesting consequence, we identify a sharp regime change in the decay of
eigenfunctions as the L\'evy intensity is varied from sub-exponential to
exponential order, and dependent on the location of the eigenvalue, in the
sense that through the transition L\'evy intensity-driven decay becomes slower
than the rate of decay of L\'evy intensity. Our approach is based on path
integration and probabilistic potential theory techniques, and all results are
also illustrated by specific examples.
