I'd say it is always dense.

Certainly, for any point $x\in M$ there is a nbd $U$ such that any $f\in W^{p,k}(M)$ supported in $U$ can be approximated by smooth functions supported in $U$ (e.g. for the same result you are quoting: we could modify $M$ in the complement of $U$ to make it compact, and nobody that lives inside $U$ would realize it).

Smooth partitions of unity do exist in any Riemannian manifold $M$. Therefore, any $f\in W^{p,k}(M)$ can be written as a locally finite sum $f=\sum_n f_n$ with $f_n\in W_c^{p,k}(U_n)$, for some locally finite open cover $\{U_n\}_n$ of $M$, by small nbd's $U_n$ were the density is true. Given $\epsilon>0$, we can choose $g_n\in C^\infty_c(U_n)$ such that $\|f_n-g_n\|_{p,k}\le 2^{-n}\epsilon$, so that the corresponding $g=\sum_n g_n$ is a smooth function with $\|f-g\|_{p,k}\le\epsilon$.