In the simple case***, both the new customer (referee) and the existing member (referrer) would either get a 1099-INT or a 1099-MISC in box 3, depending on the form of the payment.
***However, there is always room for complications if the referrer is or might be an employee or contract worker. I don't think this is what you are talking about, but I do want to emphasize that the treatment of the payment depends on your (business) relationship with the referrer.
For example, if the referrer is a regular hourly or salaried employee who collects a few bonuses because they invite friends and family, their payments are added to their paycheck as income, is subject to federal, state, social security and medicare withholding, and is included on their W-2 at the end of the year. (Even if the referrer is also a member, and even though the payments would not be job income if the referrer was only a member, they are considered wages if the referrer is a pre-existing employee. Sucks for them.)
I can also imagine an arrangement where you encourage or authorize a person to sit at a table at the shopping mall, county fair, on a busy street at rush hour, etc., to recruit and sign up new customers on your behalf. That person would be a commissioned sales person and would get a 1099-MISC with the income in box 7.
