A toothed belt would probably be fine, and would be quieter, although I don't know how it would stack up price wise.
Both the belt or chain should have a guard covering it to protect from a lost chain/belt and pinch points.
As to pulley or sprocket sizes - that math would need to be figured out once the target speed, the tire diameter, and
the motor RPM was known.
A separate motor with its own speed controller for each tire would give you the ability to put spin onto the ball to make it curve. If that isn't necessary, a single (larger) motor could be used to drive both tires.