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This could be any number of things. Take a look at the information below and see if anything resembles what is happening on your peach trees.

Trees will exhibit an oozing, clear gum-like sap when peach tree borers are present. Try scraping off the good and see if you find entrance holes and borers inside the tree. The following article discusses peach tree borers and will give you other signs to look for:
https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-borer.htm

Aphids could also be the culprit - they secrete a sticky fluid called honeydew.
https://www.gardeningknowhow.com/plant-problems/pests/insects/aphids-tree-disease.htm

It could also be bacterial canker. For more information on bacterial canker, visit the link below:
https://www.gardeningknowhow.com/plant-problems/disease/weeping-amber-in-trees.htm

Peach trees should be fertilized twice a year to help promote peach production.

Here is a link with a great article.

https://www.gardeningknowhow.com/edible/fruits/peach/fertilizing-peach-trees.htm

Brown Rot is a very common problem with some varieties of peach trees. Preventative measures is your best defense against the disease.

Here is another link with more information about dealing with the issue.

https://www.gardeningknowhow.com/edible/fruits/peach/fertilizing-peach-trees.htm

First aid is in order. Loose bark needs to removed from the girdled area - trim back to where the bark is still firmly attached to the trunk. You will then do a procedure known as "repair grafting" whereby you take healthy branches or twigs from the same tree and use them to repair the girdled trunk and restore the flow of nutrients. The repair grafting process is explained in detail here:

https://www.gardeningknowhow.com/ornamental/trees/tgen/how-to-fix-girdled-trees.htm

It can take a peach tree 2 to 4 years to produce fruit, so the fact that your tree has not produced fruit yet is not unusual.

Here is a link to peach tree care.

It has guidelines for fertilizing your tree.

https://www.gardeningknowhow.com/edible/fruits/peach/tree-with-no-peaches.htm

I believe your neighbor is correct - it is brown rot. You can read more about brown rot and other peach tree diseases, along with recommended treatments, via the link below. Basically, in order to rectify brown rot, you will need to remove all mummies from the tree and the ground, prune out cankered or dead twigs, and apply a fungicide with thiophate methyl, captan or azoxystrobin just before fruits begin to ripen.

https://www.gardeningknowhow.com/edible/fruits/peach/common-peach-diseases.htm

I think you should explore the possibility that you have a peach tree borer - it doesn't hurt to check and be on the safe side. Peach borer larvae may have found an advantageous avenue into your tree through the wound that was incurred by your lawnmower. When the borers are present trees will exhibit an oozing, clear gum-like sap. You can read more about peach tree borers and a recommended course of treatment at the following link:
https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-borer.htm

Injury, such as those sustained by lawnmowers, can also cause sap to run from the tree. For more information on repairing damaged tree bark, please visit the following link:
https://www.gardeningknowhow.com/ornamental/trees/tgen/repairing-tree-bark-damage.htm

For more information on growing peach trees, please visit the following link:
https://www.gardeningknowhow.com/edible/fruits/peach/peach-tree-care.htm