Yes putting toothpaste on cold sores does help get rid of cold sores.
The toothpaste when applied to cold sores helps to dry out the cold sore and then helps it heal faster so that the skin around the cold sore falls off naturally.
It can take a few days or so to have the cold sore fully go away with the cold sore but it will eventually work and I've done it myself with great results.
Toothpaste that contains baking soda works best but you can use other toothpastes as well.
You can also get rid of cold sores fast by taking valacyclovir (Valtrex) as soon as you first notice it developing.
It can speed up the healing process of the cold sores and make them heal much quicker.