Who is the speaker in this sentence

Question

山下さんはその映画を見ようと思う

is the speaker here Yamashita-san?

Does this sentence mean:

"Yamashita-san thinks he(Yamashita-san) will see that movie"

or

"Yamashita-san thinks I will see that movie"

Shouldn't it be 山下さんはその映画を見ようと思っている for it to mean the former?

Answer 1

Yamashita-san thinks he(Yamashita-san) will see that movie

is correct. If you want to say "He thinks I will see the movie", I think it should be

山下さんは私がその映画を見ると思っている

,

「～と思っている」 and 「～と思う」

These two are "basically" same. You can use both of these as same meaning in the first person. but, in the third person, it should be

彼は映画を見たいと思っている

not

彼は映画を見たいと思う

Sometimes, they are explained like these,

I think = 「思う」
I've been thinking = 「思っている」

but, unfortunately, 思う is sometimes used as "I have been thinking"

EDIT:
思う in the third person

A 「(あなたは)彼が何をしたいと思う？」 (What do you think he wants to do?)
B 「(私は)彼はピザを食べたいと思う」 (i think he wants to eat pizza)

There is a hidden subject 「私」
Actual subject is hidden, so it looks like that [third person]+思う. but it is not.

Answer 2

The answer: Yamashitasan is going to see the movie.

山下さんはその映画を見ようと思っている

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.

山下さんは（私が）その映画を見ると思っている

山下さんは（私が）その映画を見ようとしていると思っている

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.