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".
Again, this isn't a matter of correct or incorrect. What has been incorrect is the traditional grammar appraoch to analysing this issue.
[no slight intended to the author of this, below]
A good way to check if "I" or "me" is correct is to remove the other part of the coupling (in this case, "Tatiana") use each one by itself in the sentence:
The book was written by I.
The book was written by me.
"What is different ... is that the direct object of the verb has the form of a coordination, not a single pronoun. ...
... why should we just assume that the grammatical rules for case assignment cannot differentiate between a coordinated and a non-coordinated pronoun." [CGEL page 9]
Coordinated object pronouns used in subject position are here to stay. All the prescriptions laid upon students by teachers and parents over the years have made and will make no difference whatsoever.
The important thing to do for ESLs [and actually ENLs too], which is also what we do for other language issues is simply recognize how and where we actually use these structures. Formal and informal language differ as do speech and writing.