We find the distribution of coalescence times, birth rates, spatial velocities, and subsequent radial offsets of coalescing neutron stars (NSs) in various galactic potentials accounting for large asymmetric kicks introduced during a supernova. The birth rates of bound NS-NS binaries are quite sensitive to the magnitude of the kick velocities but are, nevertheless, similar (~ 10 per galaxy per Myr) to previous population synthesis studies. The distribution of merger times since zero-age main sequence is, however, relatively insensitive to the choice of kick velocities. With a median merger time of ~ 10^8 yr, we find that compact binaries should closely trace the star formation rate in the Universe. In a range of plausible galactic potentials (with M_galaxy3x10^10 M_solar) the median radial offset of a NS-NS merger is less than 10 kpc. At a redshift of z=1 (with H_0=65 km s^-1 Mpc^-1 and Omega=0.2), this means that half the coalescences should occur within ~1.3 arcsec from the host galaxy. In all but the most shallow potentials, 90 per cent of NS-NS binaries merge within 30 kpc of the host. We find that although the spatial distribution of coalescing neutron star binaries is consistent with the close spatial association of known optical afterglows of gamma-ray bursts (GRBs) with faint galaxies, a non-negligible fraction (~15 per cent) of GRBs should occur well outside (30 kpc) dwarf galaxy hosts. Extinction owing to dust in the host, projection of offsets, and a range in interstellar medium densities confound the true distribution of NS-NS mergers around galaxies with an observable set of optical transients/galaxy offsets.