How well would this work with public messages of people like Elon Musk or Donald Trump? Image some company training their chat-lovebots on celebs and selling them as a service. Or a creepy "friend" making a secret bot of you, and incorporating sexual content.
And a disadvantage of this will be, you can only emulate the public image of a person. It won't really contain the inner workings of a Person, and will not have the "person" grow over time.
In most scenarios that I'd get to talk to Elon or Trump, I'd probably get the public persona too. So an AI trained on that and giving it back to me isn't that useless really.
When trained on a private chat group like OPs, you get the private persona towards your friends group. You can talk to your friend for 10 years and still not really know how they would reply to their bosses email, so this isn't that much different.
And a disadvantage of this will be, you can only emulate the public image of a person. It won't really contain the inner workings of a Person, and will not have the "person" grow over time.