  We present a detailed study of the local atomic and magnetic structure of the
type-I multiferroic perovskite system (Sr,Ba)(Mn,Ti)O$_3$ using x-ray and
neutron pair distribution function (PDF) analysis, polarized neutron
scattering, and muon spin relaxation ($\mu$SR) techniques. The atomic PDF
analysis reveals widespread nanoscale tetragonal distortions of the crystal
structure even in the paraelectric phase with average cubic symmetry,
corresponding to incipient ferroelectricity in the local structure. Magnetic
PDF analysis, polarized neutron scattering, and $\mu$SR likewise confirm the
presence of short-range antiferromagnetic correlations in the paramagnetic
state, which grow in magnitude as the temperature approaches the magnetic
transition. We show that these short-range magnetic correlations coincide with
a reduction of the tetragonal (i.e. ferroelectric) distortion in the average
structure, suggesting that short-range magnetism can play an important role in
magnetoelectric and/or magnetostructural phenomena even without genuine
long-range magnetic order. The reduction of the tetragonal distortion scales
linearly with the local magnetic order parameter, pointing to spontaneous
linear magnetoelectric coupling in this system. These findings provide greater
insight into the multiferroic properties of (Sr,Ba)(Mn,Ti)O$_3$ and demonstrate
the importance of investigating the local atomic and magnetic structure to gain
a deeper understanding of the intertwined degrees of freedom in multiferroics.
