I think actors develop their characters on their own all of the time, and I believe something every actor should do. Often times the director does not have a clear idea of who a particular character is. Study the character notes in the script and how the character interacts with others. Develop your own feeling about who the character is, and then become that character on stage. Once you have completed the interpretation, the director may want to change some of the subtext you have created, and that is fine. Then you modify your interpretation of the character to match the director's interpretation. Often, a skillful, intuitive actor can develop additional dimension to a character beyond the expectations of the director, to the betterment of the production.