Pretty straight forward question.
I've seen people put it in a pitcher for between uses
I've seen people use a sanitizer solution
Do you find it gross to use it to wipe of the espresso from the drip tray?
This has been a discussion of late in our shop. You know; separating preference vs properness.
i use a hot moist towel for the steam wand only! and store on top of the machine away from anyway coffee ground material...a seperate dry towel of a different color is used for wiping out portafilters and drip tray. i also agree that the espresso machine area is to be squeaky clean ALL THE TIME!
I believe the issue is health safety and milk residue on the towel reaching warm enough temperatures to grow bacteria. This is why Starbucks has sanitizer solution - to thwart the growth of bacteria. Once bacteria begin growing then you're wiping your steam wand with a 'dirty' towel. Whatever you do, I believe changing very regularly and using exclusively for the steam wand are the keys. I also believe a sanitizer solution (that is changed regularly as well) make it that much more health safe.
Here in California everyone involved in food services of any kind had to be certified as of July 1st with a pretty thorough on-line training session that stressed the issues of cross-contamination, food safety. When it comes to dairy products it all boils down to temperature and sanitation.
Keep the cloths for exclusive steam wand use, change regularly, rinse regularly, keep in sanitizer solution - I believe those are the best practices for the optimal health safety.