  We use VLBA+VLA observations to measure the sizes of the scatter-broadened
images of 6 of the most heavily scattered known pulsars: 3 within the Galactic
Centre (GC) and 3 elsewhere in the inner Galactic plane. By combining the
measured sizes with temporal pulse broadening data from the literature and
using the thin-screen approximation, we locate the scattering medium along the
line of sight to these 6 pulsars. At least two scattering screens are needed to
explain the observations of the GC sample. We show that the screen inferred by
previous observations of SGR J1745-2900 and Sgr A*, which must be located far
from the GC, falls off in strength on scales < 0.2 degree. A second scattering
component closer to (< 2 kpc) or even (tentatively) within (< 700 pc) the GC
produces most or all of the temporal broadening observed in the other GC
pulsars. Outside the GC, the scattering locations for all three pulsars are ~2
kpc from Earth, consistent with the distance of the Carina-Sagittarius or
Scutum spiral arm. For each object the 3D scattering origin coincides with a
known HII region (and in one case also a supernova remnant), suggesting that
such objects preferentially cause the intense interstellar scattering seen
towards the Galactic plane. We show that the HII regions should contribute >
25% of the total dispersion measure (DM) towards these pulsars, and calculate
reduced DM distances. Those distances for other pulsars lying behind HII
regions may be similarly overestimated.
