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ChristianKl comments on [QUESTION]: Academic social science and machine learning - Less Wrong Discussion
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I don't have knowledge on random forests in particular but I did learn a little bit about machine learning in bioinformatics classes.
As far as I understand you can train your machine learning algorithm on one set of data and then see how it predicts values of a different set of data. That means you have values for sensitivity and specificity of your model. You can build a receiver operating characteristic (ROC) plot with it. You can also do things like seeing whether you get a different model if you build the model on a different set of your data. That can tell you whether your model is robust.
The idea of p values is to decide whether or not your model is true. In general that's not what machine learning folks are concerned with. The know that their model is a model and not reality and they care about the receiver operating characteristic.