Wow.
39 pages for such a simple concept.
There are no public areas that I could cycle in the UK without peds / dog walkers having equal rights.
On a bike, I would be expected to give way to the above (ie cycle very slowly near people and either not cycle or effectively push with 1 leg as Lee described when young kids / dogs are around).
My answer is - I ride on the road
It's really not complicated. I can't believe it's generated 39 pages of the obvious.
Clearly the law is different in the US where the car is king and no-one would consider walking or cycling somewhere a mile or so away as a sane option. (In my experience)
Hence the Jay walking laws/ hence the extra anti-cycling/ped thinking among US drivers and planners/ hence the anti non-cyclists by Digital Cowboy. They're a continuum of unpleasantness.
At least we have the rudiments of a sensible hierarchy of provision in the UK - with the most dangerous taking the most responsibility.
Of course in even more enlightened European countries this goes further, towards actual legal liability.
In the US it clearly doesn't - which is a shame.