My downstairs toilet overflowed when my washing machine drained. This happened once before and the plumber pulled debris from the clean out pipe located outside the house. I opened up the clean out plug and have pulled some debris out. There is still water in the pipe. When I plunge it, the water rises with toilet paper debris. Is there anything else I can do? Should I be able to see water in the pipe?

You shouldn't see water in the pipe at the cleanout.
This means the pipe is not draining. It will have to be augered out with a cutter as close to the size of the pipe as you can get. Check with the city if your on city sewers and see if they have a program to clean it out for you. If not a plumber is the way to go.
You can rent a snake and do it yourself but if your not sure what your doing you can run into problems and if the snake gets stuck you may be in for an expensive dig job.

It sounds like you have a blockage in the sewer line from the house to the street. This will have to be snaked. As a rule this will be roots, most likely at the street raiser. Once the roots are cleared out you can control them with RootX or Robics Foaming Root Killer that contain Dichlobrnic.
A less expensive way would be to call around to garden supply stores and ask for fine grain Copper Sulphate. Put a 1/2 pound in your toilet and flush it down. Repeat in 6 months.
Hope this helps Tom