The first area that I would start probing is at lower left, along the property line with your neighbor towards the back of the lot, and work my way out from there. Hey, I know there's one on my 1850s lot, too, but haven't found it yet. Found a well and a couple of pontils, but no privy. Yet.
Around here, they tend to be very shallow, 3 or 4 feet deep, unlined, and more than likely dipped. Hopefully, yours will be lined and full of goodies. Best of luck, my friend.