This sounds like Oriental fruit moth, a primary pest of peach that will also infest plum, apricot, nectarine, cherry, apple and pear. This pest has two or more generations of young during the growing season. The early generation larvae enter young twigs and may also feed on and enter the side of the fruit. If the fruit remains on the tree, a clear, gummy exudate will ooze from the wound, eventually turning black. Additional generations of larvae are more likely to enter nearly ripe fruit from the stem end, tunneling and feeding down to the stony pit. The pest overwinters in litter around the tree as well as on the bark; immediate removal of infested fruits and fall clean up of leaves and dropped fruit can reduce next year's woes. There has been some success of mating disruption with pheromone products as well as parasitic braconid wasps. Multipurpose fruit spray or an insecticide containing esfenvalerate or carbaryl labeled for home orchard use can provide effective control of this pest, but timing of the sprays is critical, beginning at petal fall (when 75 percent of the petals have dropped) and an additional 2-3 applications every 10-14 days. For more information on controlling pests in the home orchard, have a look at Purdue Extension Publication ID-146 Managing Pests in Home Fruit Plantings, http://extension.entm.purdue.edu/publications/ID-146.pdf
Just to add to my original response: Peach ooze is an injury response, so it could be any wound that causes the wound response. Plum curculio is another likey culprit.