Sentences (1) and (2) below use the same verb “omou”, but (1) is correct in Japanese, but (2) is incorrect.This is because the subject of sentence (1) is “I,” but the subject of sentence (2) is “my daughter.” When the verb “omou” appears at the end of a sentence, the subject must be in the first person.

(1)Watashi wa nihon de shuusyoku shiyoo to omoimasu.(I'm planning to get a job in Japan.)

(2)×Watashi no musume wa nihon de shuusyoku shiyoo to omoimasu.(My daughter is planning to get a job in Japan.)

However, when the verb is used in the form “Omotte iru”, the third person can be the subject. 

(3)Watashi no musume wa nihon de shuusyoku shiyoo to omotte imasu.(My daughter is planning to get a job in Japan.)

When using the form “Omotte iru”, the third person is not always the subject. Sometimes "I" is the subject.

（4）Watashi wa nihon de shuusyoku shiyoo to omotte iru.(I'm planning to get a job in Japan.)

So, what is the difference between “Omou”and“Omotte iru”?

　　　For example, let's say your friend invited you to a party that you wouldn't feel comfortable with.Your friend eagerly invited you, so you couldn't say no and you and your friends were heading to a party.However, along the way you changed your mind and decided not to go after all, and told your friend about it.In this situation, you should use “omou” as in sentence (5). “Omotte iru” cannot be used in this case.

(5)Watashi wa yappari iku no wo yame yoo to omou.(Actually,I'll stop going after all.)

　　However, if you withheld a reply once and you replyed again to a friend after thinking for 3 days, you will answer using “omotte iru”  as in sentence (6).

(6)Watashi wa yappri iku no wo yame yoo to omotte iru.（I'm thinking of stopping going after all.）

In other words, we use“omotte iru” because we can objectively report what we have thought about for a long time, but we use “omou” because we can state what we thought on the spot as a subjective declaration.In Japanese, what other people are thinking can only be expressed as reports.You cannot directly know what other people are thinking by looking inside their heads at the time. All you can do is observe him and guess what he's thinking.Therefore, in Japanese, it is not possible to express someone else's thoughts by saying“omou”. Please note that“omou” is a subjective declaration, and“omotte iru” is an objective report.