I stay in San Diego. The hot spot areas for spearfishing in southern California are Point Loma and La Jolla.
According to my source, only recreational spearfishing is legal in the state of California. You will have to be well aware of the restrictions regarding fishing in the protected marine areas and conserved species. Freshwater spearfishing is generally prohibited for fish species in California.

I can't exactly answer your question. After all, it's an individual thing. There are places where permission is not required, if you want you can look at Fishermap. I often find there maps where you can fish and do this kind of fishing for free.

First I think we have to know what he is calling spear fishing. Is he talking about gigging with a long handled 3 pronged barbed gig, kinda like a pitch fork or is he talking about shooting a spear from a spear gun. Two entirely different things. I don't know of any state that allows using a spear gun under water in fresh water, there may be but I doubt it.