Hi Mike,
The best way to locate the privy is with a probe. By probing every 2 ft at a slight angle. You can tell when you probe through soil that has been disturbed. Most of the yard should have a simular feel as you probe, when you come across a spot that probes much easier, check carefully to see if you can feel different layers or hear bottle crunch. The privy will have multiple use layers separated by fill layers. The fill layers will be sterile dirt with nothing in it and the use layers will have the bottles and artifacts you are looking for. The hole will probably be 3x4 and can be as much as 15ft deep in our area. If you find it you will need a digging buddy, a rope and a five gallon bucket. Once you get down to about 8 feet the bucket is the only way to get the dirt out. Also as you go down keep the hole dug out to all four sides, ( you will be able to tell the difference ) . It is very tempting to dig a smaller hole and then tunnel around the use layers , but DO NOT DO IT ... A cave in can KILL YOU. I hope this info is helpful to you ,, Good Luck and be safe..
I am up in Natchez, only a hour and a half away , if the house is 1900 or earlier and you need a hand with the dig let me know.