That's like saying: Which is more correct, "I" or "me"?
It depends on the sentence. For example:
Tatiana and I went to the theatre.
He gave the car to Tatiana and me.
In answer to the question, "Who's there?", however, opinion is divided. Some people say the correct answer is "Tatiana and I", because "Tatiana and I" is the grammatical subject. Other people say it should be "Tatiana and me", because it's not the subject but the complement, and in English the complement takes the objective pronoun.
This discussion has been going on for years, but "Tatiana and I" (in this case) is considered formal.
For this reason, you may see native speakers using "Tatiana and I" in sentences like this:
You will always be welcome to visit Tatiana and I.
This is incorrect; it should be: "Tatiana and me" in this sentence. It's an example of hypercorrection; if they have been taught formal grammar, they may have had "Tatiana and me" corrected to "Tatiana and I" on many occasions. They then believe (wrongly) that "Tatiana and me" is always wrong, and "Tatiana and I" is always correct. When they do this, they misunderstand the intentions of their teachers.
I may be showing my age here but I was always taught, and still use, "John and I" or "me and John". During my 58 years "John and I" has always, in the UK, been considered more polite and formal. In real life now, as has been said before, the majority of people use "John and me".