Do-ee is an invented word. Similar to employer and employee. The -ee ending in this case refers to the person who receives the benefit. That sentence would be better stated in this way:
"how does English allow you to tell who is doing the action and who is receiving the action?"
Yes, but I think what Dakel is saying is that "doee" is a made up word. It does not exist in English and is not found in the English dictionary. Dakel has suggested an alternative for you to use. "The person doing (or performing) the action and the person receiving the action" is a better choice.