No explanation was given as to how Maddie and Joey were able to know that Karen and Pete found out that Karen and Pete figured that Liv and Parker had been covering for them during the day while they were in LA, knowing that they'd be in a lot of trouble if they caught them enter through the door after arriving back at night.
Maddie is at the Los Angeles International airport; Diggie rushes to get there before Maddie leaves. He could have just phoned the airport and had her paged. That's what the "Courtesy Telephones" are for.
Liv thinks never to switch her identity to like Maddie and fool Karen and Pete whenever necessary (though they barely have been fooled), knowing that they would eventually need to go into the kitchen at some point during the day.
Maddie and Joey never think that Karen and Pete use the tracking feature to find their phones, which they had with them at the Airport in LA.