Combine the peeled and chopped peaches with the sugar, cover, and let macerate for 24-36 hours, stirring periodically. (I let them sit in the fridge, but you can start with 1-2 hours out on the counter to accelerate the process.)
Add the vinegars, stir, mash the fruit a bit, cover, and let sit at room temperature for a few hours to be sure the remaining sugar dissolves, stirring periodically.
Once the sugar is dissolved, mash the fruit some more, re-cover and place in the refrigerator for about a week, stirring twice a day.
Strain out the peach pieces over a clean jar, pushing down on them to release as much juice as possible. Discard the solids (or use as you see fit--they're basically pickled at this point).
Cover tightly and mark the date, so you don't forget. The shrub should be able to be stored in the refrigerator safely for several months.