
Amber F. answered 04/01/19
N1-level Japanese with Full-time Interpreter Experience
In terms of meaning, 欲しがる is used to talk about the state or attitude of wanting something. It's a bit of a subtle difference, but 欲しがる is more like to hanker for, to yearn for, to have a craving for.
Grammatically, the difference between 欲しい and 欲しがる is that 欲しい is an adjective and 欲しがる is a verb, so the way they are used in sentences will be different.
So you can say,
けいこさんが優しいです。
Keiko-san = kind (です, which you can leave out in casual speech, acts like the equals sign)
or
(私は)ポルシェが欲しいです。
Porsche = desired (and the 私は、which you can state or leave out, means, "by the way, this sentence is describing my feelings").
If you want to use 欲しがる、 the sentence will look like this:
(私は)ポルシェを欲しがっています。
(My) state or being or attitude is one of wanting a Porsche.
In terms of politeness level, there is no difference between the two. For both the adjective 欲しい and the verb 欲しがる、 you have to end the sentence with です or ます if you want the sentence to be in the normal-polite form.
(私は)ポルシェが欲しいです。(normal-polite)
(私は)ポルシェが欲しい。 (casual)
(私は)ポルシェを欲しがっています。 (normal-polite)
(私は)ポルシェを欲しがっている。 (casual)
I have never heard anything like what you are saying about only being able to use 欲しがる when the subject is not present in the room. But normally you do not go around saying to people, "You crave ______, don't you?" so in most situations you probably wouldn't tell someone they are 欲しがっている unless it is a casual relationship. And language that is polite (like when you get into 敬語, humble and honorific language) is absolutely language that you can use when talking to the person in question.
For your example sentence, you would need to put in *who* should be eating the vegetables in order for the sentence to be correct.
先生は野菜を食べて欲しがっていますか?
先生は、私が野菜を食べてほしいです。 Sensei wants me to eat vegetables.
先生は、私が野菜を食べて欲しがっています。 Sensei is yearning for me to eat vegetables.
In this case, if you omit *who* is eating the vegetables, the reader/listener is going to assume that it is probably 先生、but if you want to say 先生 wants to eat vegetables, you wouldn't put it this way, you would just say, 先生は野菜が食べたいです。
Finally, if you wanted to ask your teacher this question, as you can probably see from the translations above, "Sensei is yearning for me to eat vegetables" is a little bit of a strange thing to say, so it would be better to just ask, 先生は、私が野菜を食べてほしいですか? (I would also put an お in front of 野菜, especially if you are female.)