Potty training is probably the messiest experience you'll go through when your kids are little. Even a kiddie who seems like they've mastered it can have accidents, and a parent needs to be prepared for every possibility. The key to this is planning... and stuffing that diaper bag as full as possible for every anticipated need. We did our initial potty training days at home, with no pants, (the kids, not me). Eventually, though, we needed to venture out and having the right items in the diaper bag when potty training was key.
If you thought your bag was filled when your baby was using endless diapers, be prepared for a whole new set of "must-haves" while they are phasing out of them. When my nephew was segueing to using the toilet, we went on a road trip from New York to Michigan. I wish I could say it was a great success and that he peed and pooped in the potty with perfection, but it would be more accurate to say that the trip was filled with more poop than you can imagine — from clean ups in Aisle 3 of the supermarket to all over the shag carpet of our rental house. From this, we learned the value of a fully stocked diaper bag with as many changes of clothing as possible.
Like everything in parenting, it's all about being prepared.