How do they tell if I had bacterial or viral bronchitis without further testing?

You can't. Viral or bacterial bronchitis produce much the same symptoms. Bacterial may produce a more productive cough but not always.
Clinically... Typically, a bacterial infection has more severe symptoms than a viral infection. For example, there is usually a higher fever, more sputum production, etc. Your doctor may be able to determine this by examining you. Sometimes, further testing, like a sputum culture, may be necessary but often the diagnosis can be made by your doctor on clinical grounds alone.