Tina, different teachers charge very different rates for dancing with students at the competition. Her are the various situations I've heard of over the years:
Flat Fee Per Dance -- This is simple and straightforward, and is the most common fee structure in my experience. I've heard rates of between $20 and $60 per dance. If the competition involves travel, the teachers travel expenses are evenly split between all the students who dance with that teacher.
Flat Fee Per Competition -- This is also simple and straightforward. I've heard rates of between $100 and $10,000 per competition. If the flat fee is on the high end of the scale, it sometimes includes travel.
Sliding Scale Fee Per Dance -- Teachers who are trying to get their students to do a lot of entries sometimes do this. I've heard of one teacher who would charge $20 per dance, but the more entries you did the lower the per-dance fee would slide until it hit about $8 per dance.
Year-Based Retainer Fee -- This is not common in the USA, but I've heard of Pro/Am dancers in Hong Kong who do this. The student pays the teacher a retainer fee to do a certain number of competitions per year. This amount can be very very high, as there seems to be a lot of money in the Hong Kong scene. The most well-known case of this was in the New York Times this summer, where a woman paid her teacher something like $8million US for a multi-year Pro/Am contract.
In my personal opinion, I think the flat fee per dance + students evenly share travel expenses is the clearest and most fair. One of my teachers used to present me with an itemized bill for each competition, listing our entry fees, our tickets to the ballroom, his per-dance fee, and my share of his travel expenses (which were split between all his students who would dance with him at the out-of-town competitions). Based on my experience, paying your teacher $2k - $3k for a one day competition seems like a LOT, unless you are dancing with one of the absolute top pros in the country. If you're dancing with a former or current US finalist (or world representative), it could cost you as these teachers have the skill and reputation to pretty much guarantee getting their students to the top of Pro/Am. I know one such top-of-the-top teacher who charges a flat fee of $5k per competition. And his students have been the best over the years, winning national Pro/Am championships, so they are certainly getting what they are paying for.
By the way, my first Pro/Am competition was between two or three local studios, held one afternoon at one of the studios, and cost me about $150. More recently, I was paying entry fees + ballroom tickets + share of travel expenses + $40 per dance, which worked out to be about $650 for a large sanctioned local competition (so without travel expenses).