First of all, your horsey friends are WRONG about rain rot being caused by rain. IT ISN'T !!! Rain rot, or rain scald as it is called in Europe, is caused by a fungus which lives in the soil. The horses get it by rolling in the soil- and it's possible for a horse to get rain rot WITHOUT ever being exposed to rain and moisture. I should know- I have had to treat and work with horses which were infected with it, even in places where it was very dry, such as in California during the dry season.
As for the other part of your question, yes, it is VERY possible to transmit the fungus from horse to horse by sharing brushes, blankets, saddle pads, and any other personal item you can think of. Rain rot is HIGHLY contagious. We currently have 2 horses on our farm which have this condition, and they both share the same pasture- but not the same brushes or blankets. There is also another horse, in the field next door, who has the condition- and HE doesn't share ANYTHING with the first two- not even a water trough, and yet he also has this problem. From what I just described, it should be rather obvious to you that rain rot is caused by more than rainfall exposure.
You are the one who is right in this argument, and so are the authors of the books you have been reading. Rain rot is treated in a variety of ways, but the best treatment I know of is iodine ( betadine) in baby oil. The oil loosens the scabs, and the iodine will kill the fungus and allow them to heal. This is what we are using with our horses right now, and it works well.
I hope this helps settle the arguement.