Long story short: I've been running a start-up company for a couple years, and we're now making the jump to a larger, funded operation. The current plan calls for hiring a producer, but it's not a role I'm overly familiar with and so I'm wondering if you guys feel there's some particular 'bang for the buck' team size for a producer. Especially for a dedicated producer We're probably going to be around 6-7 engineering staff and 3 artist/designer, where one of those designers might be doubling as producer. Is that shared responsibility a bad idea at our target size?

I'd say for any team size greater than say half a dozen, having a producer is important -- dedicated, not so much. It detracts from work having to do 'producer chores' and having an individual dedicated to the roll can be a big help to keep focus on content creation instead of business management duties, but for a small team that might prove an overburden since budgets tend to be tighter and any kind of overhead should be avoided if at all possible. However, if you have someone experienced enough already in the team (s)he could take on the roll of producer - just be aware that it will affect how much work that can get done when wearing more than one hat.

If you are less than a dozen I'd probably say 'yes' to a shared responsibility and not forcing on a dedicated producer. However, a dozen or more is definitely the time to start getting someone to handle the producer tasks separately - now, whether you find someone internally to do the job or look for someone externally to the job is completely up to you and if anyone is readily available and up to the task.

Good luck and good hunting!

We're probably going to be around 6-7 engineering staff and 3 artist/designer, where one of those designers might be doubling as producer. Is that shared responsibility a bad idea at our target size?

A producer's main job is to facilitate - to make sure that everybody has what they need so they can do their jobs. He should also be looking ahead to anticipate needs. And watching for red flags, and playing Whack-a-Flag on them as soon as they pop up. With a small team, there may not be full-time producing work to do, so he could do additional duties in those quiet interludes. But the project can't count on him for that doubled-up duty full time, because he has to be able to jump immediately when producer work has to be done.
I recommend a couple of books about producing: The Game Production Handbook (Chandler) and Introduction to Game Development (Rabin). Full disclosure: I wrote the production chapter in Rabin's book, and Chandler worked with me at Activision.