Interestingly, native speakers can put themselves first, but for it to be natural you have to make other grammatical errors!
For example,
My sister and I disagree with you. (Correct)
Me and my sister disagree with you. (Not correct, but very natural and native, with a slightly different meaning. To me, it suggests that my disagreement might be more important than hers. But changing ‘me’ to ‘I’ here makes the sentence less correct, not more! You can’t do it.)