Applying toothpaste to the cold sore won't help it go away overnight or very quickly but the toothpaste when applied to the cold sore can help dry out the cold sore and help it go away and fall off much sooner.
So toothpaste does work to get rid of a cold sore because I've tried it myself but it took 3 days to get rid of the cold sore with toothpaste.
Another home remedy for getting rid of a cold sore is to take some salt and water and dip a Q Tip in the salt water and apply the salt water to your cold sore.
You can also apply baking soda and water to the cold sore which helps it go away.