Using Twitter to spread the status of a web application

I am creating a web application that many people will use at the same time. I have a requirement that whenever a user has a significant event (for example, a win in a game), information about it is sent to all other users. There is nothing for them to do about it, just to let them see a list of interesting things that are happening. I assume that these significant events will occur at a speed of about two minutes per minute for the entire system.

I thought about all the server resources and code that would be needed to support this, and then I suddenly thought that I could just do it using Twitter. Each time a significant event occurred, our server could send a message about this, and each page of the web application may include a small javascript widget displaying all these tweets. As an added bonus, people who just wanted to keep track of what was happening with the application could do this with a simple Twitter client.

Before I delved into this, I wanted to hear some feedback on whether you think this is a good idea or not. What things should I look at to do it right?