Let $\mathfrak{M}\_g$ denote the moduli stack of Riemann surfaces of genus $g$, it is a smooth complex analytic stack, and is the analytic stack underlying $\mathsf{M}\_g$, the moduli stack of complex algebraic curves of genus $g$. Alternatively, it is the quotient stack $\mathcal{T}\_g // \Gamma\_g$ of the mapping class group acting on Teichmuller space (by biholomorphisms).


There are two things we might mean by the Picard group of $\mathfrak{M}\_g$, using either holomorphic line bundles or algebraic line bundles. The first is $Pic\_{hol} := H^1(\mathfrak{M}\_g;\mathcal{O}^\times\_{\mathfrak{M}\_g})$ and the second is $Pic\_{alg} := H^1(\mathsf{M}\_g;\mathcal{O}^\times\_{\mathsf{M}\_g})$. As $\mathfrak{M}\_g$ has a finite cover by the quasiprojective variety $\mathfrak{M}\_g[3]$ of curves with a level 3 structure, I believe the latter group can also be defined to be the subgroup of $Pic\_{hol}$ of those holomorphic line bundles which become algebraic on $\mathfrak{M}\_g[3]$. If not, this defines yet another notion of Picard group.


It seems to me that it is $Pic\_{alg}$ which is usually discussed, for example in Mumford's paper on Picard groups of moduli problems. Is $Pic\_{hol}$ discussed explicitly anywhere? On the other hand, presumably these groups agree (certainly $Pic\_{alg} = \mathbb{Z}$ is a summand of $Pic\_{hol}$). Is this explicitly stated and proved anywhere?


Any other observations about this distinction are welcome too.


EDIT: As a related question, suppose that one only knew about the analytic object $\mathfrak{M}\_g$. Can one directly show that $c\_1 : Pic\_{hol} \to H^2(\mathfrak{M}\_g;\mathbb{Z})$ is injective? (Granted $H^1(\mathfrak{M}\_g;\mathbb{Q})=0$, say.)



Picard group of $\mathfrak{M}_g$
This will be a bit sketchy, but hopefully you can fill in the necessary steps.
If you see a problem, let me know.


To avoid getting distracted with stacks, let me use a level $n\ge 3$ structure, and
write $M=M\_{g}[n]$. Let $j:M\hookrightarrow \bar M$ be the Satake (not Deligne-Mumford) compactification.
This is the normalization of the closure of the image of $M$ in the Satake compactification
of $A\_{g}[n]$. The point is that $codim (\bar M-M)\ge 2$. So given a holomorphic line 
bundle $L$ on $M$, $j\_\*L$ would be a coherent analytic sheaf. Therefore by GAGA
$j\_\*L$ is coherent algebraic, so that $L$ is an algebraic line bundle. This
implies surjectivity of map $Pic\_{alg}\to Pic\_{hol}$ in your notation. 


This should finish it, since as you said,
you already know injectivity of the above map.


