If someone get cold sores occasionally, and uses their chapstick(most times applying it to the finger...?

then putting it on the jaws instead of rubbing the tube on their lips) and sometimes applying it with the tube not the finger...can they spread the virus if they shared their chapstick stuff next to someone else? In other words...someone I know gets cold sores on the oral cavity, and they sometimes share their lip gloss/chapstick with my daughter...I'm worried that my daughter may expiration up with cold sores from sharing chapstick...she one and only wants what she see the other person doing...and the individual that has the cold sores know not to share stuff when they have a cold sore...but can't they still spread the virus through chapstick even if they don't hold one visible? Yes, it's possible even if sores aren't distinct.