Upper roof downspout emptying onto lower roof
We have a similar two story house where we had contractors redo the gutters.  They continued the downspout from the second floor over the roof of the first floor and positioned it directly into the hole in the first floor gutter. 

No straps were visible on the run over the first floor and it appears to be secured at the second floor downspout only.

Volume of water has not been given a real test yet.  I direct the output into rain barrels with an overflow facing downhill. You could direct it into 4" perforated pipe with sleeve buried in the ground and running away from the house. This is a French drain which empties into a dry well.  If you ever wondered how to use up empty plastic pails this is your chance!