Good question. Smart tvs are essentially computers. They have their own internal hard drives and memory but it's all self-managed.
The movie itself is stored in the cloud. The only thing that might be left on your computer are cookies and perhaps the software used to access the movies. I stream movies and TV show to my TV and the TV's built in software take care of the management of but the movies are not stored on your PC as that would be illegal so you do not have to worry about it.
You're right that the movie itself is stored in the cloud, but as mentioned previously in this thread, when you stream the movie to your computer, chunks of the movie are downloaded to the computer in order to create a buffer in case the internet speed degrades. If you are streaming the movie using your browser, you'll see new files added into the browser's cache folder as you watch the movie. The easiest way to test that chunks of the movie are indeed downloaded to your computer while streaming is by disconnecting the internet from your computer while you are streaming. You'll notice that the video still plays for a while even with no internet connection.