You definitely want to separate DTOs per endpoint into 2 gruops: request and response ones.
Now, speaking of inheritance, you could have base Request and base Response class which would include some general, non-specific things (for example, request&response wrapper message into which you plugin your data) but you don't want to mix inheritance between request and response side.
As others already pointed out, it is a bit more code at the begining but that way you'll never hit the wall. On a contrary, mixing them will pretty soon make you have huge refactorings where you'll lose more time&nerves than starting in a clear, separate way.
