The main thing we should be asking ourselves is whether knowing when somebody else is typing will really have a significant enough impact on our behavior.
Because that's really what the feature is for on other sites that have it—when someone else is typing, especially in a heated conversation, it helps to know that they are so you can stop for a moment and let them finish. It's somewhat like an actual conversation. Is it polite or appropriate to interrupt someone when you're speaking with them? Absolutely not.
And, perhaps, waiting for the next message (because I know lots of people that submit sentence fragments, or submit incomplete thoughts with intents to complete it in their next message) could better influence or guide what your next message may be.
So the answer to the question of whether it will have an impact is dependent on the current behavior and habits of our population. From what I, personally, have observed, most chat users do submit complete thoughts in each message. Such is a benefit of the more sophisticated, grammar-conscious community that the SE sites are able to maintain.
But we must refer back to the purpose of the chats, which is really to discuss something that either takes up too much space on the actual site or just isn't a proper fit for the Q&A system. The other day, I wanted a professional opinion on a design I was doing, so I went over to the UX.SE chat and asked for it. I knew I couldn't mold it into the Q&A model that the SE sites rely on for their core content, so I decided to use the chat system instead.
So, I can't speak on behalf of the entire chat population, but given the purpose of the chat system, or at least what I perceive it to be, this feature would be an excellent addition.
However, given the habits of most of the users in chat, it would be more appropriate in the context of Stack Exchange and likely cause less confusion to not make this kind of addition to the chat system.
So my final stance on this, with the vision of SE and the chat system in mind, is no.
EDIT: I say no especially because I see no practical implementation of this. You're absolutely right, a text indicator would be extremely confusing, but an icon over a user's avatar in the sidebar would be so hidden and inconvenient. When I'm typing I either look at what I'm typing or at the last few messages, never at the sidebar.