The fact that there are multiple transporter rooms aboard Picard's E-D was never really in dispute: basically every time they used a transporter room, they specified its number ("Room 3" was over-represented), and for example "11001001" included dialogue about a very large number of transporter rooms, up to "transporter twenty".
However, the transporter rooms of Kirk's ship were never mentioned in plural or given numbers. So we have to go for speculation and deduction in this case (which is closer to what the original poster was interested in).
If you go strictly by on-screen canon, there are some contradictory signs, but never anything conclusive. There is the repeated reference by Kirk and others to "the transporter room" generically, but never anything conclusive that it was singular or plural in terms of the overall ship's equipment.
Now, Stephen Whitfield & Gene Roddenberry were conclusive in "The Making of Star Trek" in '68. They specified multiple six-man transporter rooms, as well as 22-man and cargo facilities as well.