Localized turbulent structures such as puffs or oblique stripes are building blocks of the intermittency regimes in subcritical wall-bounded shear flows. These turbulent structures are investigated in incompressible pressure-driven annular pipe flow using direct numerical simulations in long domains. For low enough radius ratio η, these coherent structures have a dynamics comparable to that of puffs in cylindrical pipe flow. For η larger than 0.5, they take the shape of helical stripes inclined with respect to the axial direction. The transition from puffs to stripes is analyzed statistically by focusing on the axisymmetry properties of the associated large-scale flows. It is shown that the transition is gradual: as the azimuthal confinement relaxes, allowing for an azimuthal large-scale component, oblique stripes emerge as predicted in the planar limit. The generality of this transition mechanism is discussed in the context of subcritical shear flows.