In a Pro-Am competition, a professional partner (Pro) competes with an amateur dance (Am). The amateur dancer is the only one evaluated by the judges; the professional in this case is on the dance floor to facilitate the amateur (either by leading or following) but is not actually judged.
In a Pro-Am, the amateur is responsible for all financial costs of competing, including registration. In addition, the amateur pays their professional partner to compete with them during the actual event. More info about budgeting and Pro-Am competition costs can be found here.
In an Am-Am competition, two amateur dancers compete together against other amateur couples. Both the leader and follower are evaluated, and their final score is based on how each one of them performs, both individually and together. Generally, amateur couples split registration and other costs (like coaching and lessons) equally.
However, depending on circumstances one amateur may pay more or less than the other. There is no right or wrong way to split up costs. What’s important is that you and your amateur partner openly discuss financial commitments and expectations before you start competing or practicing together to make sure that you are on the same page. If your financial circumstances change after you have started competing together, make sure to talk to your partner as soon as possible, so that the two of you can re-set together. Like any other relationship, strong dance partnerships are developed through open, honest, and consistent communication.