We present radial surface brightness profiles for all five globular clusters in the Fornax dwarf spheroidal galaxy, and for the four present members of the Sagittarius dwarf spheroidal galaxy. These profiles are derived from archival Hubble Space Telescope observations, and have been calculated using the same techniques with which we measured profiles in our previous studies of LMC and SMC clusters (astro-ph/0209031 and astro-ph/0209046 respectively), apart from some small modifications. From the surface brightness profiles, we have determined structural parameters for each cluster, including core radii and luminosity and mass estimates. We also provide a brief summary of literature measurements of other parameters for these clusters, including their ages, metallicities and distances. Our core radius measurements are mostly in good agreement with those from previous lower resolution studies, although for several clusters our new values are significantly different. The profile for Fornax cluster 5 does not appear to be well fit by a King-type model and we suggest that it is a post core-collapse candidate. We examine the distribution of cluster core radii in each of the two dwarf galaxy systems, and compare these with the distribution of core radii for old LMC clusters. The three distributions match within the limits of measurement errors and the small sample sizes. We discuss the implications of this in the context of the radius-age trend we have previously highlighted for the Magellanic Cloud clusters.