My almost 2 year old daughter recently started telling family ,friends and even strangers to "Go Away." If they don't, she gets angry, yells and even pushes people. I've tried asking her why (she is very verbal) with no luck. I've told her that if she wants to be by herself then she should play in another part of they room where I can still see her but that I (and others) are not going away. She's probably demonstrating independence, but I don't know how to handle this.
I would not be too upset by your daughter's telling everyone to "Go away!"--even people she does not know. A child of almost two is often out-of-sorts, and trying to control other people and push them around somewhat is not unusual.
You are right that she is demonstrating independence--but she is still so little that she really doesn't know what she really wants to be independent from; a minute later, she is likely to demand that you come close all over again.
I would try to handle her demands with as much respect as possible. You can probably show her that her wishes have been heard, at least most of the time, by backing off for a few feet and for a few minutes. Little children live in the moment--she is not asking you to book an ocean voyage, she just wants you out of her hair right now. If she feels successful in getting you to acknowledge her feelings, she will probably calm down in a bit and move on. You may be in for several weeks of this. When she gains more experience with getting the world to cooperate with her wishes, she will grow less impatient and more diplomatic. But at this time, she may have a short fuse. Of course, you want to help her learn that pushing people physically is not OK--you can intervene while you explain, "No no, we don't push."
Two year olds are just learning how to argue--they even argue with themselves! But trying to discuss this is probably over her head. I'd try to go with the flow calmly, as best you can, and to show her that you understand she just needs a little space.
Elizabeth Berger MD
Child Psychiatrist and author of "Raising Kids with Character"