  We study leptogenesis in three different realisations of the type Ib seesaw
mechanism, where the effective masses of the neutrinos are obtained by the
spontaneous symmetry breaking of two different Higgs doublets. In the minimal
type Ib seesaw model, where two right-handed neutrinos form a Dirac mass, we
show that it is impossible to produce the correct baryon asymmetry, even
including a pseudo-Dirac mass splitting. In an extended type Ib seesaw model,
with a third very heavy Majorana right-handed neutrino, together with the low
scale Dirac pair of right-handed (RH) neutrinos and an extra singlet Higgs
boson, we find that the imbalance of matter and antimatter can be explained by
resonant leptogenesis. In the resulting low scale effective type Ib seesaw
mechanism, we derive the allowed range of the seesaw couplings consistent with
resonant leptogenesis. Dark matter may also be included via the right-handed
neutrino portal. The Dirac RH neutrino masses may lie in the 1-100 GeV mass
range, accessible to the future experiments SHiP and FCC-$ee$, allowing the
type Ib seesaw mechanism with leptogenesis and dark matter to be tested.
