http://phonl.com/fl_law/rules/FRCP/frcp1310.htm
"(4)
Any deposition may be recorded by videotape without leave of the court or stipulation of the parties, provided the deposition is taken in accordance with this subdivision.
(A) Notice. A party intending to videotape a deposition shall state in the notice that the deposition is to be videotaped and shall give the name and address of the operator.
(B) Stenographer. Videotaped depositions shall also be recorded stenographically, unless all parties agree otherwise.
(C) Procedure. At the beginning of the deposition, the officer before whom it is taken shall, on camera: (i) identify the style of the action, (ii) state the date, and (iii) swear the witness.
(D) Custody of Tape and Copies. The attorney for the party requesting the videotaping of the deposition shall take custody of and be responsible for the safeguarding of the videotape, shall permit the viewing of it by the opposing party, and, if requested, shall provide a copy of the videotape at the expense of the party requesting the copy.
(E) Cost of Videotaped Depositions. The party requesting the videotaping shall bear the initial cost of videotaping."
Looks like they can't stop the videotaping of the depos.
I haven't found anything yet that addresses the media/public attendance at depos. The law in Florida is turning out to be quite interesting.