Incidentally, as I posted this question someone who knew the answer wandered into my office.

The map $M_g \to A_g$ factors through the moduli space $\tau_g$ of pairs (A,P,L) where A is an abelian variety, P is an A torsor, and L is a translation invariant line bundle on P. The map $M_g \to \tau_g$ is given by $C \mapsto (Pic_0, Pic_{g-1}, L(\theta))$, where the theta divisor on $Pic_{g-1}$ is given by the image of $C^{g-1}$.

To construct the map $\tau_g \to A_g$, note that $Pic_0(A) \cong \Pic_0(P)$, so that L indeed gives a map $A \to A^{\vee}$ given by $a \mapsto t^*_aL \otimes L^{-1}$.

The point is one doesn't need to descent the actual theta divisor. The reference to this is 2.1 of Martin Olsson's book Compactifying moduli spaces of abelian varieties.

Incidentally, as I posted this question someone who knew the answer wandered into my office.

The map $M_g \to A_g$ factors through the moduli space $\tau_g$ of pairs (A,P,L) where A is an abelian variety, P is an A torsor, and L is an ample line bundle on P which is geometrically a principal polarization. The map $M_g \to \tau_g$ is given by $C \mapsto (Pic_0, Pic_{g-1}, L(\theta))$, where the theta divisor on $Pic_{g-1}$ is given by the image of $C^{g-1}$.

To construct the map $\tau_g \to A_g$, note that $Pic_0(A) \cong Pic_0(P)$, so that L indeed gives a map $A \to A^{\vee}$ given by $a \mapsto t^*_aL \otimes L^{-1}$.

The point is one doesn't need to descend the theta divisor. The reference to this is 5.1 of Martin Olsson's book Compactifying moduli spaces of abelian varieties.