What do you mean by "production of the deposition"?
If you are suggesting that the OP can cause to be "produced" for her personal use a transcript of her oral deposition by any means other than making arrangements with the reporter who took and pay for it, you are mistaken.
Seemingly you are confusing transcripts and/or electronic recordings of depositions as "documents, writings, tangible things, etc." , " that can be obtained, inspected and copied - or procured by subpoena duces tecum. But such are not with the scope of Rules 1.350 or Rule 1.351 or any other.
If the OP wants a transcript of her deposition, she must arrange for it through the reporter that took it and pay for it.
Moreover, it would be unbefitting and a marked disservice to the reporter in subverting his or her means of livelihood for the opposing lawyer to provide her a duplication of his copy.