Burn't charcoal or wood is an excellent alternative to toothpaste.And the best part is that there's no fluoride in the ashes.
I don't think it's really healthy to digest burnt charcoal, there must be carcinogens caused by burning. A friend (dentist student) told me, that his teacher told students them that no toothpaste is needed, and only mechanicaly remove plaque helps and toothpaste has no effect, except you don't "dry" brush your teeth, plus it's easier to teach kids to brush their teeth when you use flavoured tpaste. I believe it, because I can see what shit is between my teeth when using dental floss and I don't think any tpaste would help me remove it. I still use tpaste though, probably for better breath