Technically, yes. You can say "leggo, scrivi" and be understood, because the verbs are conjugated to imply who reads, and who writes. Further, you could simply write "Leggo." and this would be grammatically and semantically correct. Obviously you could do the same with "scrivi". Think about it: "you write/you are writing" is a perfectly acceptable sentence in English and is the direct translation.

Italian is what is known as a "pro-drop" language which means, as Jess mentioned, that the pronoun can be omitted because conjugated verbs conform to the first, second and third person.

As for the sentence being "awkward," I agree it does sound a bit abrupt. You wouldn't really use it in conversation unless you wanted to send someone on their way.

As I understand it, the subject pronoun (io, tu etc) can always be omitted if it is clear from the verb or from the context who is the subject of the verb. In this case the verb endings makes it perfectly clear who is the subject of each verb, so omission of io and tu is fine.

(Duolingo accepted "leggo, scrivi" when I put it in)

Absolutely! Pronouns can always be omitted in Italian, because the conjugation of the verb signifies which tense is stressed.

No. "Leggo, scrivi" has nosense and no one speak or write so. Here we need tu specify subjects: "Io leggo, tu scrivi"

I think it could. But since it is such a short sentence without any nouns, I do not know if there is another rule? You could say: leggo libro or scrivi libro. Can a native explain it maybe?

Yeah it accepted it for me too. But I want to be as exact with traditional Italian. It's one thing to translate colloquial English into Italian, it's another to speak real, accurate Italian. So, is it accurate to say "leggo, scrivi"?

As a native Portuguese speaker, this would look awkward in my language, so I assumed the same thing would happen in Italian.

Tu is singular voi is plural

Lei has two meanings, she and you (formal mode) voi just means you (plural)

In Italian, Lei with a capital L is the formal version of you whereas tu is more informal. You'd use Lei when speaking to your boss or teacher etc also if you don't know someone well then you'd use Lei at first until you became closer.

No significa lo mismo. Pasa igual que en el español: tu siendo "tu" en singular y no formal, mientras que voi es "ustedes" en plural. Pasa que en ingles you significa tu y ustedes por igual.