  We evaluate the contribution of pion-pion rescattering to the $p p \to p p
\pi^+ \pi^-$ and $p \bar p \to p \bar p \pi^+ \pi^-$ reactions. We compare our
results with the close-to-threshold experimental data. The pion-pion
rescattering contribution is found there to be negligible. The predictions for
future experiments with PANDA detector at High Energy Storage Ring (HESR) in
GSI Darmstadt are presented. It is discussed how to cut off the dominant Roper
resonance and double-$\Delta$ excitation mechanisms leading to the $p p \pi^+
\pi^-$ channel in final state. Differential distributions in momentum
transfers, transverse momentum, effective two-pions mass, relative azimuthal
angle between pions, and pion rapidities are presented.
