The Set Designer Is responsible for creating an appropriate performing space for the actors, both from the viewpoint of functionality (having suitable entrances as required by the script or director), and from the artistic appearance of the set (the decor/style). Some directors will have a very strong view about the overall design, whereas others will give their designers a greater input into creating the look of the show.
For best results, the Set Designer should work with the other designers to ensure the set design is consistent with the overall design ideas. A production will look a lot better if each designer is following a consistent thread and taking into account of the look and atmosphere being created by the other designers.
Shows in Cambridge vary as to whether the Set Designer also takes responsibility for the building and painting of the set, or if these jobs are organised by a separate Technical Director and/or Master Carpenter. You should discuss in early production meetings exactly where the boundaries of the Set Designer role lie and what you are prepared to do for any given production, so as to ensure that people can be added to the production team in these other roles as soon as possible, if they will be required.
For larger shows, the set designer shouldn't attempt to paint everything! Instead, it's common to appoint set painters. This can cover more advanced painting of landscapes and complex backdrops, or simpler tasks such as painting items of set.