Let me start by saying that the vast majority of people are just fine using whatever they grab of off the shelf. However, there are a fair number of chemicals added to most toothpastes today and a small percentage of people will have issues to some of those ingredients. One ingredient in particular that causes problems is SLS, or sodium lauryl sulfate. SLS is a detergent and is in toothpastes in order to make it foam up. Unfortunately some people are sensitive to it and will get burning, tingling, or red gums from using it. Usually it only happens when you mix it with harsh alcohol containing mouthwashes and some peroxides to whiten. It is a combination of the alcohol, SLS and peroxide that cause issues the majority of the time, but any of them could by themselves.
There are several great SLS free toothpastes out there. Three of the most popular are Tom’s of Maine, Biotene, and KMF. All of these brands have different products that vary in how “natural” they really are so check to see if SLS is listed in the ingredients.
I know some people do not want fluoride in their toothpaste and I believe you can even find some that are fluoride free. However, unless you are eating a diet that is 100% consistent with our hunter/gather ancestors you will get decay. If you rely on grains (preferably with some grit like sand to help clean the teeth), fruits and vegetables, and meat for 100% of your diet and are not eating anything processed, canned, or packaged in anyway you might be OK. Realistically that diet is nearly impossible, so unless you like spending time and money with us dentists, use fluoride in your toothpaste.