We calculate the structure and neutron content of neutrino-heated magnetohydrodynamic winds driven from the surface of newly-formed magnetars ( “ proto-magnetars ” ) and from the midplane of hyper-accreting disks , two of the possible central engines for gamma-ray bursts ( GRBs ) and hyper-energetic supernovae ( SNe ) . Both the surface of proto-magnetars and the midplane of neutrino-cooled accretion flows ( NDAFs ) are electron degenerate and neutron-rich ( neutron-to-proton ratio n / p \gg 1 ) . If this substantial free neutron excess is preserved to large radii in ultra-relativistic outflows , several important observational consequences may result . Weak interaction processes , however , can drive n / p to \sim 1 in the nondegenerate regions that obtain just above the surfaces of NDAFs and proto-magnetars . Our calculations show that mildly relativistic ( Lorentz factor \Gamma \lesssim 10 ) neutron-rich outflows from NDAFs are possible in the presence of a strong poloidal magnetic field . However , neutron-rich winds possess a minimum mass-loss rate that likely precludes simultaneously neutron-rich and ultra-relativistic ( \Gamma \gtrsim 100 ) NDAF winds accompanying a substantial accretion power . In contrast , proto-magnetars are capable of producing neutron-rich long-duration GRB outflows \sim 10 - 30 seconds following core bounce for sub-millisecond rotation periods ; such outflows would , however , accompany only extremely energetic events , in which the GRB + SN energy budget exceeds \sim 4 \times 10 ^ { 52 } ergs . Neutron-rich highly relativistic outflows may also be produced during some short -duration GRBs by geometrically thick accretion disks formed from compact object mergers . The implications for r -process nucleosynthesis , optical transients due to non-relativistic neutron-rich winds , and Nickel production in proto-magnetar and NDAF winds are also briefly discussed .