No. Performing Rights Organizations such as ASCAP and BMI collect performance royalties worldwide, but there are many other types of publishing royalties they don't collect. For example, they do NOT collect mechanical royalties. CD Baby Pro can collect your mechanical earnings for you.
When you register your songs with ASCAP or BMI, they share their data with foreign PROs so you can get paid for foreign performances. When you sign up for CD Baby Pro, we'll ensure that your P.R.O. song registrations are correctly listed all over the world.