A properly formed sentence with the structure either ... or conjoins two equivalent things. For example, they could be two adjectives that modify the same noun, two clauses that fill the same role in the sentence, or two nouns that are both complements of the same preposition.
That said, quite often people conjoin two items after "either" that do not follow this rule, yet the intended meaning is so clear that there's no doubt what the intended meaning is, and it sounds like good grammar. This is one of few situations in English where technically bad grammar sounds just fine. In writing, an editor would notice this and correct it. In speaking, nobody would even bat an eyelash.
- ... that I either am interested in or have read before.
In this example, the conjoined items are both verb phrases, so the grammar is good.
- ... that I am either interested in or have read before.
In this example, the first conjoined item is an adjective, while the second is a verb phrase, so technically it's wrong, but most people hearing or reading it would understand it perfectly and not think it was a mistake.
- ... that either I am interested in _____ or I have read ______ before.
Here, both items after "either" are clauses with a missing object, which is represented by the relative pronoun "that". The grammar is fine.
In English, we have a strong distaste for repetition, and it sounds bad anywhere it can be avoided. In this third sentence, "I" is repeated unnecessarily. Since sentence 1. has correct grammar and is the simplest way to use the "either ... or" structure, it's the best choice.
BUT 1. is not the most natural. "Either" is optional where the meaning is still clear without it. So a natural phrasing of this would be:
Below are the articles that I am interested in or have read before. (sentence 1. without "either")