A director already has a huge responsibility to handle creating the visual story for an audience. Why hinder that responsibility with producing details? If I were a director, I would have a producer at my side at all times!! I would find that one person who gets me and build a lifelong relationship that carries me through most projects. Sure, I may go and work with other producers at times but I would find that one I can develop my own ideas with and build a mutually beneficial and creative partnership for the long haul. But that's just me. Each director may have different desires or needs, but each should consider having a separate producer who can handle all or the lion's share of the producing on a film.
A producer and director relationship is like a marriage. They are the ones in charge of the success of the film, both creatively and financially. They build the family on that film and make many decisions together. It's inevitable that arguments may occur but like any strong marriage, compromise is essential. Technically, a producer has final say but that should be reserved for only major last-resort situations. To be honest, I have never had to use my final say with any of the directors with whom I have worked as I have always tried to be diplomatic and compromise as have the directors.
When choosing a producer or director, try to find someone you can see yourself being friends with and who shares similar values on top of having a shared vision for the project. You will be spending years working together, day in, day out. You may even see him or her more than your spouse or partner. So make sure you like that person!
Looking back at the directors I have worked with, they all share common traits of being calm, intelligent, creative, and open to the opinion of others (while never compromising the quality of the film or their vision). And I love and respect each and every one and continue to work with them all to this day. I cherish my relationships with my directors. And I believe they feel the same. We lean on each other in this crazy industry and I know we help each other stay sane.
So I guess I have a "type" of director I like working with. Figure out the "type" of producer or director you like and search for him or her. Find a partner to share the load. I think you will find your work to be even more enjoyable and fulfilling.