...the two words are confused when writing them down.

[sunsunmoon]

Be careful not to write the word of when you mean have or 've: I could've told you that not I could of told you that. The mistake arises from the fact that the pronunciation of have in unstressed contexts is the same as that of of, and the two words are confused when writing them down.
(From having - WordReference.com Dictionary of English)

Are there any words left out after "...are confused when"? Has "you are" been omitted?

[emsr2d2]

Be careful not to write the word of when you mean have or 've: I could've told you that not I could of told you that. The mistake arises from the fact that the pronunciation of have in unstressed contexts is the same as that of of, and the two words are confused when writing them down.
(From having - WordReference.com Dictionary of English)

Are there any words left out after "...are confused when"? Has "you are" been omitted?

The words that have been omitted are "with each other". However, they are not necessary.

The two words are frequently confused [with each other].

The original is fine.

[sunsunmoon]

...and the two words are confused when writing them down.

Could I ask if there are any words omitted between when and writing?

[emsr2d2]

...and the two words are confused when writing them down.

Could I ask if there are any words omitted between when and writing?

No, there aren't any words missing. It could be reworded as "...the two words are confused when people write them down" but the original is fine.