I would peel the spots off off, but that's just based on my own opinion. I have absolutely no idea if they are safe to eat, but regularly take off 'bad spots' from veggies and eat the rest.

Those spots are just cosmetic. I have never read anywhere that they were bad for you. Peeling the skin does reduce the health benefits of peaches. Almost all of the homegrown peaches will have some spots. If you are worried you could wash the peaches with a fruit wash such as Fit vegetable and fruit wash. I try to wash all of my store bought fruit, mainly for contaminates from people that have touch it before I bought it.