Is it the other guy’s responsibility to set the expectation on when to get to it?

If it is stopping you, then talk to your supervisor/boss/manager/whatever. It’s literally their job to “unstuck” you, especially when it’s due to differing business priorities (i.e. red tape).

As another comment has mentioned, if it truly is a priority than at least with digital communication you’ve established a paper trail. In person confrontations diminish the length and quality of this trail.