How do I unclog underground downspout pipes?
I ran into the same situation on two of my own houses.  I knew where the exits were, but they were long runs, with extended, packed blockage (probably never cleared).  Initial attempts at clearing them with plumbing snakes and special hose nozzles didn't make much of a dent.  

You've got nothing to lose trying to clear the pipes.  But if they have many years of build-up that has turned into a giant solid plug, and efforts to clear them don't seem to get anywhere, it may not be worth the time to clear them. That's especially true if you don't own serious pipe cleanout equipment.  You can get them professionally reamed out, but that isn't cheap.

At one house, I just dug up the buried pipe and replaced it.  If you're careful, you can put the dirt and sod back.  That took about the same amount of time that I had already spent trying to clear them, and was cheaper than getting them professionally cleared.

If you attempt to clear the pipe with a snake or hose attachment, can locate the plug and determine how long it is, it might save a little work to cut out and replace that section if it is at one end of the pipe.  

At the other house, I abandoned it in place and just used downspout extenders.