It's a sign of the times that most of the discussion on public sector job relocation has been about the impact on the public sector finances. When the Lyon's report considered the issue in 2004 it highlighted cost savings, the efficiency of delivery and the impact on regional economies.
It is surprising how little we know about the last of these. The evidence considered by the Lyon's review comes in four forms. Some case study evidence, some multiplier analysis (where you use information on expenditure patterns to figure out how one job generates more jobs as local suppliers of goods and services react to increased local demand), some theoretical modelling and some correlations between local private employment and local public sector employment. I am not sure that I find any of this particularly convincing and am currently trying to figure out how to get at the effects in a more systematic manner.
I suspect, however, that the lack of any definitive evidence in either direction will not stop people presenting speculation about the effects as fact.