I will just develop a bit the comment by Jason Starr. In these notes
http://math.stanford.edu/~conrad/papers/coarsespace.pdf
you can find the following theorem by Keel and Mori:

Let $S$ be a scheme and let $\mathcal{X}$ be an Artin stack that is locally of finite presentation over $S$ and has ﬁnite inertia stack $I_S(\mathcal{X})$. There exists a coarse moduli space $\pi:\mathcal{X}\rightarrow X$, such that:

- The structure map $X\rightarrow S$ is separated if $\mathcal{X}\rightarrow S$ is separated, and it is locally of finite type if $S$ is locally noetherian.
- The map $\pi$ is proper and quasi-ﬁnite.

Moreover, if $X^{'}\rightarrow X$ is a ﬂat map of algebraic spaces then $\pi:\mathcal{X}^{'}:=\mathcal{X}\times_{X}X^{'}\rightarrow X^{'}$ is a coarse moduli space.

In you case the map $Spec(\overline{K})\rightarrow Spec(K)$ is flat. Therefore $\overline{M}_{g,n}^{K}\times_{Spec(K)}Spec(\overline{K})$ is a coarse moduli space for the stack $\overline{\mathcal{M}}_{g,n}^{\overline{K}}$. Since the coarse moduli space of $\overline{\mathcal{M}}_{g,n}^{\overline{K}}$ is unique up to a unique isomorphism you get $\overline{M}_{g,n}^{K}\times_{Spec(K)}Spec(\overline{K})\cong \overline{M}_{g,n}^{\overline{K}}$.

setof $K$-valued points of the scheme $\overline{M}_{g,n}$, not a (finite type) $K$-scheme. You could ask, is the natural morphism of $\overline{K}$-schemes, $$\overline{M}_{g,n,\overline{K}} \to \overline{M}_{g,n,K}\times_{\text{Spec}(K)} \text{Spec}(\overline{K})$$ an isomorphism? The answer to this is yes, because the formation of the coarse moduli space is compatible with flat base change. I recommend reading Keel-Mori for a detailed discussion of this property. $\endgroup$