I used to work for an enterprise social company in 2011. I happen to find a folder of resumes for product manager and interview feedback--on a common windows share. And this enterprise social company is not top tier at: all those candidates had MBAs from US schools. Even this company was selective wrt which MBA one had: Kellog vs Cornell Johnson. What this tells me is this: the bay area market has too many MBAs (morning program, evening program, weekend program, etc). So, companies are selective when they got lots of candidates. GE Software in San Ramon is a good place to look for a product manager role. Their pay is not that great (no RSUs, just salary + bonus): it is a bad place to work for some people. But it is a great place for other kinds of folks. GE has too many CIO's, CTO's, Senior Directors, Directors, etc, compared to other companies: so, many guys called "GE boys" play all games to move up. They got too many leadership programs too.
You need to be local for these jobs or at least you need to have someone push your candidacy for the job, then come onsite for interviews.
Are they promoting product managers within? . Usually, Intuit has such a program wherein a fresh college grad can move from software engineer role to associate product manager role, etc. So, some companies do promote internally.