Convergence in $C^{\infty}_{\text{loc}}$ on a manifold

Question

Usually on a domain in $\Omega \subset \mathbb R^n$, convergence in $C^{\infty}_{\text{loc}}(\Omega)$ just means that for each compact subset $K$ of $\Omega$ we have $C^{\infty}(K)$ convergence. However that does not seem to be plausible on a manifold because any compact set on a manifold might not be a manifold. So what is the formal definition of $C^{\infty}_{\text{loc}}(M)$ convergence on a manifold $M$. Do we require smooth convergence on each compact submanifold?