The idea that cosmic relativistic jets are magnetically driven Poynting-dominated flows has many attractive features but also some problems. One of them is the low efficiency of shock dissipation in highly magnetized plasma. Indeed, the observations of gamma-ray bursts (GRBs) and their afterglow emission indicate very high radiative efficiency of relativistic jets associated with these phenomena. We have revisited the issue of shock dissipation and emission and its implications for the internal shock model of the prompt GRB emission and studied it in the context of impulsive Poynting-dominated flows. Our results show that unless the magnetization of GRB jets is extremely high, σ > 100 in the prompt emission zone, the magnetic model may still be compatible with the observations. First, for σ≃ 1 the dissipation efficiency of fast magnetosonic shock is still quite high, ∼30 per cent. Secondly, the main effect of reduced dissipation efficiency is merely an increase in the size of the dissipation zone, and even for highly magnetized GRB jets, this size may remain below the external shock radius, provided the central engine can emit magnetic shells on the time-scale well below the typical observed variability scale of 1 s. Our analytical and numerical results suggest that strong interaction between shells begins not during the coasting phase but well before it. As the result, the impulsive jet in the dissipation zone is best described not as a collection of shells but as a continuous highly magnetized flow with a high amplitude magnetosonic wave component. How exactly the dissipated wave energy is distributed between the radiation and the bulk kinetic energy of radial jets depends on the relative rates of radiative and adiabatic cooling. In the fast radiative cooling regime, the corresponding radiative efficiency can be as high as the wave contribution to their energy budget, independently of the magnetization. Moreover, after leaving the zone of prompt emission, the jet may still remain Poynting dominated, leading to weaker emission from the reverse shock compared to non-magnetic models. Energetically subdominant weakly magnetized ‘clouds’ in otherwise strongly magnetized jets may significantly increase the overall efficiency of the shock dissipation.