Can anyone tell me how it is handled if a passenger either catches a cold, has a stomach flu, etc and is unable to participate in tours for a day or two but doesn't want to end the trip entirely?

Is it possible to just stay on the bus? Or would the driver make arrangements to take someone back to the hotel early? Or would those kind of details fall on the passenger?

Myself and travel companion are prone to getting a "bug" while traveling but not long lasting enough to want to end the tour early -- but also not wanting to subject others to our illness.

options?

We have been on several tours where people have been sick, or had injuries. Most people with colds and minor stomach problems just get on the bus in the morning and go with the rest of the tour. Yes, colds are passed around on the bus. In some cases the director is notified and he/she will arrange for a doctor or hospital visit. Yes, you can say on the bus when the driver also stays. If the driver leaves, the bus is locked and no one is allowed to stay in our experience. At most stops there will be a safe place for you to sit comfortably while the tour group goes site seeing. If you decide to stay in bed and tour moves on to another location for the night, you will have to find your own way to catch up. We watched this in Switzerland. The director provided instructions to the passengers concerned and they caught up by train the next day.