You can redirect the stdout / stderr (standary out put / error stream) of a process if you are the one starting it. For an example take a look at this.
Capturing the output stream of a process which was not started by you, well, that is whole different matter. I'm not sure it can be done.
But if you have control over the source code of both apps, there are other ways to communicate, like pipes / remoting / WCF, and so on...
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