You will brush and then spit out the toothpaste materials, so chemicals are not of great issue. Not all toothpaste contain fluorides, nor does they need it. If not ingested fluorides are of no danger and they will make the teeth resistant to dental carries. The main ingredient of a toothpaste is abrasives such as calcium carbonate which is also available as medicine. Preservatives are the one everybody is so cautious of nowadays but those are present in medicine like syrup as well.
So, you are okay. Even if you ingest it, the damage is usually not of great concern. Since you won't eat it, no need to worry about the harmful chemicals.