Here is how I clean the stones (when I'm working in spa where I'm pressed for time)
Lay a towel on the massage table with used linens on it. Lay out the stones in ordered rows on this towel. It doesn't matter how many stones are in each row, just that they are not in a jumble. I have a spray bottle with alcohol from the drugstore.
Spray one row of stones thoroughly with the alcohol. Quickly use a hand towel to rub them down, before the alcohol dries. Then do the next row. This is why it's important to keep them in rows -- so you know which ones have been cleaned and which haven't.
When you finish all rows (I usually have 4-5 rows) go through and systematically turn over each stone. You now have to clean the other side! Repeat alcohol scrub on other side, row by row.
Then go to your stone holder, whatever you use, and clean that out. Spray alcohol liberally in the container that no longer has any water in it, then wipe down with the hand towel.
If it is a roaster, and you are able to keep the main heater on while you clean these things, that is good. If you are able to then fill up the heater with hot water from the facuet you are even better off. In those 5-7 minutes (that's all it should take you once you get this method down), your stones have not lost their core heat, so you'll be good for the next session.
At home, when I schedule my own appointments, I scrub each stone in the sink with a dishwashing sponge and soap. Same with the stone container/heater. But when you're in a spa and they give you two stone massages back to back, sometimes you have to do the best with what you're given. In which case I use the above method.
ETA: only use this quick method with store bought polished stones that have very little texture and don't hold much dirt. If you are using hand picked stones that are most likely rough and pick up skin and dirt more easily, ONLY USE THE SOAP AND SINK METHOD.