What is the relationship like be

This has more to do with the process and culture of an organization than anything specific to the job title of designer or product manager, in my opinion.

When PMs and PDs work together from the beginning and share a common understanding of the goals that this product/feature is supposed to achieve, it's an energizing collaborative working relationship that turns out great useful, usable, delightful product.

But when that isn't the case, here's some of what you hear:

PMs: I want it done faster so we can get in more features.

PDs: I want to do one more revision so it's closer to perfect.

PMs: Why does PD care so much about these stupid details?

PDs: Why are PMs okay with all these inconsistencies when the user testing we did shows that people are confused?

PMs:  This design doesn't meet our business goals.

PDs:  No one ever tells us what the business goals are, but everyone has an opinion about what colors the damn buttons are.

PMs:  Customers aren't going to buy us because of our design.

PDs:  But customers will complain/file bugs/stop using us because of our lack of design.

There's often a fracture line between the minds of product managers and designers because:

When taking decisions, PMs tend to be data-driven and crave reliability, whereas designers tend to be inspiration-driven and look for validity.

The way designers work tends to irritate PMs because most of what designers apparently do in the course of a design project looks to untrained eyes like a waste of time (everything that is not actually pushing pixels or vectors looks like a waste of time to non-design-savvy people).

On the other hand, too many designers happen to be poor time managers and  actually tend to spend the first two thirds of the project slacking off if left unmonitored.

Too many PMs do a terribly poor job of committing to specific, focused statements on target public, business objectives, and positioning. They just want "everything for everybody" and are unable to renounce to features / target groups / attributes in order to gain positioning sharpness.

Too many designers look driven by personal likes/dislikes or by their self-expression needs instead of by the project's actual objectives and time/resource constraints.

Too many PMs lack elemental knowledge about the main principles and heuristics of design, and have zero design critique and appreciation skills. As a result, they fail to articulate feedback on designers' work in terms more serious than "I don't like that color" or "I don't quite see the wow factor".

Too many designers have serious ego issues and take any criticism to their work as a personal attack.

Too many PMs tend to micromanage design decisions while failing to provide strategic direction to the design process.

Too many designers have an excess of style. And, like Charles Eames used to say, "a designer has a style to the extent that he fails to understand the design problem".