The answer: Yamashitasan is going to see the movie.
You suspected it was also able to mean "Yamashita-san thinks I will see that movie" because we can cut subjects off sentences in Japanese, didn't you? It is, however, beyond the possibility this time. There is no reason to assume the another subject exists, and you must not do it on this sentence.
If you want to say "Yamashita-san thinks I will see that movie" in Japanese, "山下さんはその映画を見ようと思っている" is not enough because "(subject) + その映画を見よう" is not full-sentence, it needs "と思っている" or some.
"(verb) + よう" tells us the subject's will, "going to (verb)" in English, which you know that you need a "be-verb" before. What you need in both languages are alike.
You also need "と" and "～する," "～思う" or something for "その映画を見よう," and that works as a "be-verb." And you know that the subject of "be-verb" and "(verb)" is exactly the same one. In Japanese, too.
When you say so, it means "Yamashita-san thinks I will see that movie." As you see, boldface ones are clearly full-sentences, "I will see that movie." You can finally presume there is another subject in the sentence which conteins another sencence in itself.
And I wonder how you picked up exactly "I" from the Japanese sentence. Anyone or anything can be subject if it was omitted, I think.