Normally but it depends upon the volume.
For a building, or significant part thereof, then heavy rain will cause a lot of water in a short space of time and a soakaway is needed to cope with that volume. For a small (very small?) structure it may be sufficient to run the water into a section of surface drain that just drains into the ground, possibly in more than one place.
Pergolas come in many sizes (and some do not have roofs or only partial ones) so you would need to think about how much water could fall on it in heavy rain.
Another option (again for something small) would be to run the water into a water-butt and take the overflow from that away onto the garden in soaker hose. You might need a section of ordinary hose to take the water across (more likely under/alongside) any patio between the butt and the start of the garden proper.