I estimate that I get a BoE stone every 100 battles or so where I am fighting versus level 25 wild pets. After looking at my total number of battles won, that's my best estimate.
Today I happened to get one right before logging off, after a streak of not finding any in a while. A few days ago I got one on my second fight, so it varies ultimately, and just purely my data alone isn't enough to give a detailed enough picture of the true drop rates.
I have yet to get a BoP stone from wild battles so far, which I find extremely odd, as I've easily gotten 10-15 BoE's.
All of my BoP stones came from daily bags. For those getting BoPs from wild fights: are you fighting versus 24s/25s? Or lower? I typically only battle versus 25s and sometimes rarely 24s if they happen to also be around. I'm wondering if this is the difference in obtaining the BoE versus a class-specific BoP stone.
I don't really go out seeking to find stones, and not stones of a particular type. Instead, my goal is to level enough pets to 25 to get the Singing Cricket pet. Only 5 more to go. The battle-stones I get along the way are just extra surprises.