  We consider radio bursts that originate from extragalactic neutron stars
(NSs) by addressing three questions about source distances. What are the
physical limitations on coherent radiation at GHz frequencies? Do they permit
detection at cosmological distances? How many bursts per NS are needed to
produce the inferred burst rate $\sim 10^3$-$10^4 $sky$^{-1}$ day$^{-1}$? The
burst rate is comparable to the NS formation rate in a Hubble volume, requiring
only one per NS if they are bright enough. However, radiation physics causes us
to favor a closer population. More bursts per NS are then required but repeats
in 10 to 100 yr could still be negligible. Bursts are modeled as sub-ns,
coherent shot pulses superposed incoherently to produce ms-duration $\sim 1$ Jy
amplitudes; each shot-pulse can be much weaker than the burst amplitude,
placing less restrictive requirements on the emission process. Nonetheless,
single shot pulses are similar to the extreme, unresolved ($< 0.4$ ns) MJy shot
pulse seen from the Crab pulsar, which is consistent with coherent curvature
radiation emitted near the light cylinder by an almost neutral clump with net
charge $\sim \pm 10^{21}e$ and total energy $\gtrsim 10^{23}$ ergs. Bursts from
Gpc distances require incoherent superposition of $\sim 10^{12}d_{\rm Gpc}^2$
shot pulses or a total energy $\gtrsim 10^{35} d_{\rm Gpc}^2$ erg. The energy
reservoir near the light cylinder limits the detection distance to $\lesssim
{\rm few} \times 100$ Mpc for a fluence $\sim 1$ Jy ms unless conditions are
more extreme than for the Crab pulsar. Similarly, extreme single pulses from
ordinary pulsars and magnetars could be detectable from throughout the Local
Group and perhaps farther. Contributions to dispersion measures from galaxy
clusters will be significant for some of the bursts. We discuss tests for the
signatures of bursts associated with extragalactic NSs.
