The online portion secures your boarding position only-and does require you to enter PP info. Without your PP handy, you cannot even do this part. Actual "check in" occurs at the airport. You cannot get a boarding pass for an international flight anywhere but the airport. Below is the link from southwest.com

The OP had an issue with some travelers where (most likely) info on the PP didn't match the reservation exactly. In those cases, you have to do everything at the airport. Usually, that is the middle name-or naming convention.

https://www.southwest.com/html/air/i...-in/index.html