The aim of the present research is the analysis of the photon motion in the regular spacetimes arising as solutions of the Einstein gravity coupled with a non-linear electrodynamics (NED). The photons no longer follow the null geodesic of the background spacetime, but the null geodesics of an effective geometry where the electromagnetic non-linearity is directly reflected in addition to the spacetime geometry. Motion of photons is compared to the motion of neutrinos that are not directly affected by the non-linearities of a non-Maxwellian electromagnetic field, and follow null geodesics of the background spacetime. We determine shadows of the regular Bardeen black holes, representing a special solution of the general relativity coupled with NED related to a magnetic charge, both for photons and neutrinos, and compare them to the shadow of the related Reissner-Nordstrom black holes. We demonstrate that the direct NED effects give clear signature of the presence of the regular black holes, on the level going up to 20% that is detectable by recent observational techniques. We also demonstrate strong influence of the NED effects on deflection angle of photons moving in the Bardeen spacetimes, and on the time delay of the motion of photons and neutrinos in vicinity of the black hole horizon.