I can't see the data on another computer in my network.
Why not?
I have the webduino server running together with my application
I have no idea what this means, since you can only have one sketch at a time running. That is either "your application" (whatever that means) or the "webduino server". Perhaps you mean that you modified the webduino server sketch to do additional stuff. In which case, where's the code?
So my question is about a possibility to output data like over the serial monitor, but over the ethernet to another machine.
A server can't push data. The client needs to contact the server and request the data. The server can return a page with meta tags that causes the client to do another GET after a specified period of time.