I’m talking about the kind found in OTC skin treatments. Will it bleach (or stain?) clothes?
Yes. Whenever I’ve used acne washes and treatments with salicylic acid, my hand towels and pillowcases have become bleached. I would then use only old towels for my face. I just had a neighbor remark that her daughter’s towels were losing color and bleaching. I told her it was due to salicylic acid products and she should switch to white towels and pillowcases. I have never had a problem with clothes, but I would think if one were using the products on the body, it would affect clothes in the same way. 
Yes. Like wombattver, I lost a lot of nice towels and washcloths when my daughter was using salicylic acid. She lost a few pairs of pants by wiping her hands across her rear.
Thank you.