It seems to me that there is an important distinction between the cases where the hands play synchronously or asynchronously. In the former instance the hands share an implied commonality of metre or even notation, perhaps complicated but still shared, as in a Bach fugue, a rag or even most jazz improvisation. In asynchronous playing the hands seem to operate like two different people.
I have come to prefer this latter way in improvisation lately, but when all is said and done, both ways are just options, each to be used as appropriate. Synchronous playing just happens to suit written notation whereas most asynchronous playing cannot be clearly notated at all. For me the whole process is intimately tied up with accents and the fact that some notes are more important than others to the ear and the mind. This is true of any music, not just improvisation. If it were not so then, however complex the contrapuntal relationship between the hands, a piece would be a perfectly flat collection of notes with no musical interest and no life.
A more general form of your question is how to allocate hand use to musical impulse during improvisation. I have to admit that recently I increasingly make both hands do duty all over the place as required to execute my flow of ideas. So while independence of hands probably exists physically, it is not strictly true to say that they are ever independent musically. If that effect were desired it would be easier to play two tape recordings at once.
It's a good question, and one which has far deeper creative implication than its simplicity would suggest. In the liner notes of Jarrett's "Radiance" he asserts that he is just starting to listen to things his left hand has to say. Coming from an improviser of his age and experience this extraordinary admission testifies to the complexity of the relationship, mental and physical, between the hands during improvisation.
Topic: hand independence