DISCLAIMER: This answer does not provide references, nor is a formally thought-out proof.

Only some (hopefully) useful heuristics.

You begin with a topos E with associated theory T(E).

From T one creates a locale L (and a localic groupoid G(L) ), such that there is a surjection Sh(L)----> E, where Sh(L) is the topos of ALL sheaves on L. The topos E is then recovered by isolating the equivariant sheaves, ie G(L) -equivariants. Now, let T(Sh(L)) the theory of the topos Sh(L).

The question is basically: *how T(sh(L)) relates to the original T(E)*.

Conjecture: there is some kind of modality on T(sh(L)) such that T(E) is gotten as the "fixed points" for that modality.

The intuition behind this conjecture is that the topos E is obtained by considering only well-behaved sheaves in the pool of all sheaves. They are, in a sense, the ones which are invariant with respect to some shuffling of the topos sh(L) by some suitable action.

NOTE: My (unpublished) dissertation had a somewhat germane theme: what I was after was some "toposophical" semantics of general modal logics, and the trick was to consider topoi endowed with an extra lex endofunctor. The endofunctor was then used to isolate a fixed-points subcat (or more generally a cat of coalgebras for the endo) which had the nice property that the subobject functor had a built-in modal operator (induced by the endo). I do not know if the Joyal -Tierney representation produces something along similar lines, but the guess is yes.