Integrating screen sharing into a JavaScript video chat app enhances collaboration by allowing users to share their screens during calls, improving communication clarity and facilitating interactive learning experiences. To implement this feature using VideoSDK, developers must first create an account and obtain an auth token from the dashboard. They should then install VideoSDK in their project and structure it accordingly. Next, they need to design the user interface (UI), implement join screen functionality, initialize the meeting, create media elements, handle participant events, and implement controls. Finally, developers can enable screen sharing with or without audio by following specific steps provided by VideoSDK.