How can I get this circuit to "make noise"
As other people have mentioned, you are going to have to amplify the output.

The speaker probably pulled the output voltage to near zero. If you want to verify this, try measuring the output while the speaker is connected to the output. You should find the white noise is gone. 

Can use another transistor to amplify, or a simple op amp follower or amplifier.

Or you could do something like this if you want to use a specific chip: http://www.circuitbasics.com/build-a-great-sounding-audio-amplifier-with-bass-boost-from-the-lm386/

Quick tip though, use a crystal piezoelectric earphone. It should have high impedance so you can plug it directly into your circuit without affecting its behavior too much.