In the UK roof water usually goes to soakaways, and I think private driveways will often do so too.
However, there is generally only one public drainage system, so if it's going to a proper pipe, it'll be the public sewer. More houses off the public drainage will have a septic tank and will keep rainwater out of it.
Road drainage in the countryside usually goes into local surface drains and ditches. Newer ones will have oil skimmers and perhaps balance ponds (those square pits you see beside motorways) to cope with sudden storms.
Current best practice uses porous materials ("Grasscrete" etc) for driveways and residential access roads, so most rainwater soaks straight in.
I think some places in the world have a parallel "grey water" drainage system for baths, roofs etc, with a "black water" sewer for proper sewage.