It is secretive because many studios do not wish you to know how much the studio and teacher are actually making off of you. Plus, there generally is no discount for teachers taking multiple students. Also, remember that the studio needs to make its money during the competition. And instructors won't do competitions week in and week out unless they are making good money.
I believe that all competitions now post the prices online if students decide to attend without an instructor or studio. Which means that the studio will not let you join them if they have paying students with them. But if you choose to take up the time of an instructor or studio, then they will charge you what they feel is appropriate.
Let's look at something simple as if you were the owner of the studio.
$100 / entry to competition x 2 = $200
$25 per heat for competition and $15 per heat for teacher = $40 x 20 heats = $800
$300 per day for teacher ($200) and studio ($100) x 7 days = $2100
$10 / breakfast, $15 / lunch, $20 / dinner = $45 x 2 x 7 days = $630
$500 / flight x 2 = $1000
$150 / night x 2 = $2100
$200 & $800 & $2100 & $630 & $1000 & $2100 = $6830 - $2400 for studio/instructor = $4430 towards competion and expenses. So the studio and instructor really aren't making much off of you.
What am I missing? Costumes, shoes, makeup, hair, jewelry, lessons.
Instructor made $200 for the 20 heats
Instructor made $1400 for the daily rate for taking care of you for over 14 hours per day
Instructor $1400 & $200 = $1600 / 7 = $230 per day essentially
Studio made $100 for the 20 heats
Studio made $700 for the daily rate of taking away their instructor
Studio $100 & $700 = $800 / 7 = $110 per day essentially.
Studio $100 / hour for private lessons x 6 hrs teaching per day = $600 - $180 instructors pay = $420. A studio is losing money by sending a student to a competition unless they can get multiple students per instructor.
I hope that helps.