My dd (5) does the same thing with the private, made-up language. It is quite clever and complex when she gets going. I've always thought she was very creative, imaginative, and intelligent to do that, so I don't want to stifle it or her. That said, I find it terribly annoying when she talks to me in it and expects me to understand her. I guess some part of me feels it is disrespectful to talk to someone in a language they do not share. I think the response on my part may be somewhat irrational. After all, she is probably just trying to get me to play a game with her, join her in her world, etc. I find that very difficult. Lots of moms on here talk about playing with their children, but I don't do it very well. I do try and am learning, but find that many of the other adults she is close to, do it gladly and willingly so I cut myself some slack. Anyway, I am off an a tangent! What I do is explain to dd that I think she is very funny, clever, etc. and it is good to use her imagination, but that I don't like it and not everyone likes everything, so she can play those games with people who enjoy them, or when she is playing by herself. I do sometimes hear her dolls and toys "talking" to each other in a made-up language.
I don't know if that helps. It is kind of a different situation, but maybe your dd would be able to differentiate times when she can use her language in games and play, and times when she needs to communicate with you in a language you both understand. Of course, you would probably have to talk to her about it when she is not in the middle of a tantrum!