I designed this noise generator circuit that uses a 1N514 Schottky diode with a VDB transistor amplifier(note: the upper VDB resistor is actually 15kOhm). Looking at the oscilloscope trace it seems to work fine in generating continuous white noise with a fairly flat FFT spectrum and maximum peaks between 4 and 5 volts. But when I try to connect the output to a small speaker I don't hear anything. I haven't done much frequency analysis but perhaps should I add a frequency shifter or low-pass filter so the output is in the audible range?
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.