Can a California P.I. Take Somebody's Trash?

Yes.

In fact, anybody can take somebody's trash in California as long as it's not on private property. On trash day when the bin is at the curb, it's not on private property. Phone records, bank statements, credit card bills, and private mail are all fair game if discovered in the trash.

A 1988 California Supreme Court ruling stated that leaving trash in an area “particularly suited for public inspection” nullifies an expectation of privacy because, “it is common knowledge that plastic garbage bags left along a public street are readily accessible to animals, children, scavengers, snoops, and other members of the public.”

However, this doesn't apply to recycling bins! 

Once recycle bins are off private property, they become property of local government agencies, so to speak. So it's illegal to "steal" from designated recycling bins under California law.

Unfortunately, the recycling bin is more likely to contain the records or other paper evidence I'd be interested in for an investigation. But the regular trash can provide access to other types of discarded items that could be useful evidence or help confirm suspicions.