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No, bating them wouldn't do this, but going to the shelter sure would.
They most likely have an upper respiratory infection (URI) or kitty herpes. Both are very contagious from one cat to another and could have been picked very easily in an animal shelter.
Can they get rid of it themselves? Maybe and maybe not.
I would try and treat them, and then see if show improvement in 72 hours or not. Here is what you can do at home.
Keep the nose and eyes clean of discharge. You can use warm water dampened cotton balls or Qtips to do this. You can also bring them in the bathroom when you shower. This will allow the steam to help break any congestion in the head and chest. A vaporizer can well too for this.
Give them each a 1/4 tablet of chlor-trimeton a day. This will help runny nose, sneezing, and watery eyes.
If the eyes have a green or yellow mucus like discharge, that means there is infection in the eye. If so get some Terramycin
I am not going to kid you, I do think they need a vet to help get over this. They need some clavamox. You can only get his from a vet as it is a prescription antibiotic. It is not expensive either. Many times if URI are left untreated they can cause they can cause eye ulcers or even pneumonia.
Please let me know if you need further assistance with this.