Can betta fish transmit diseases to other fish in the same bowl?

Yes. All fish can transmit certain diseases to all other fish.

Some diseases are species-specific but, like humans, sometimes one disease will lower the immune system, and give the fish other illnesses as well.

If you notice your fish is diseased or dying, Quarintine it in a seperate tank/bowl, with the highest quality foods and keep a close eye on pH, Ammonia and Nitrate levels.