"What's the problem, Jon? This sort of thing happens all the time."
"It does?"
"Sure, remember the accident last year outside the pizza place where that guy split up into twins? Shelly and Ralph, their names were. There's nothing special about your transporter."
"Karyn," said Jon, "I have no idea what you're... Oh! That must be it. If transporters are now a real part of the world, the world must also have procedures in place to deal with transporter accidents. We're not the first people this ever happened to."
"Of course we're not. Let's go to City Hall and get this new girl registered." Karyn looked at the girl and asked "What's your name?"
"I don't think I have one," she answered. Jon realized--she didn't say her name was Karyn, recognize Jon or Karyn, or otherwise act like she thought she was Karyn. She knew how to do things like talk, but she didn't know anything.
"Well, how about 'Jill'," said Karyn.
"I like that name," said Jill.
"And I wish Jon would step back into the transporter, and it would correctly fix him back to himself because somehow we found a DNA sample for him after all."
Jon stepped into the transporter and it hummed. A few seconds later he was out again, and back to normal.
After getting Jill some clothes, Karyn took Jon and Jill to City Hall, where she filled out a long form as Jon and Jill watched.
"What's this?" asked Jon. "Parents... Jon and Karyn?"
"She came from a part of you and DNA from me. That makes us her parents."
"It does? But we haven't even... Sigh. What's this age test thing?"
"Well, depending on how a transporter breaks down if someone comes into existence they might have an exact copy of someone's mind, or they might just know things without memories like Jill, or they might know nothing at all. They could have different mental ages."
"I want to take the test," said Jill.
The employee behind the window looked up some numbers and wrote down a number on a slip of paper, which she handed to Jill. "Computer's over there," she said. "Use this login."
Jon and Karyn watched Jill take the test, but Karyn shushed Jon when he tried to give hints. Eventually Jill finished and the results came up--teenage. Which wasn't surprising, really.
So the next day at school the teacher introduced the new student. Named Jill, she looked just like Karyn and her official parents were Jon and Karyn.... Of course, in a world where transporters exist everyone knew about transporter accidents, so while they thought it was unusual they didn't think it was of earth-shattering weirdness.
"You know," said Jon, "if transporters exist, and transporter malfunctions really exist, could we intentionally mess up a transporter to pull pranks on other people?"
"I don't know," said Karyn. "You're thinking of turning Sarah into a dog or a guy or something?"
"Maybe not Sarah, but it's the same idea. I couldn't do that with the stone because I didn't want to permanently change someone, but transporter accidents can be fixed."
"But that's mean!" said Karyn. "Hmm. Let me think..."