Usually the voltage drop on a diode is considered relative low noise. For the white noise part the differential resistance is relevant parameter. For the ideal diode this is kT/e / I - and thus independent of the diode. Real diodes usually have a slightly higher resistance. So already at 1 mA the resistance is in the 50 Ohms range and thus a white noise level of only 1 nV/Sqrt(Hz) is expected. In the high frequency region, so that the diode is considered slow, the differential resistance is even lower - thus slow PIN diodes can be even lower noise above some 100 kHz.
There might be some additional 1/f or similar low frequency noise contribution. Just from my feeling and picture of how a diode works this is expected to be higher in diodes that are slow and have a higher forward voltage / low leakage. Not much of this is expected for fast gold doped diodes like the 1N4148.
Current noise, for the reverse leakage is a different thing. Here charge quatisation gives a major noise contribution, thus again diode type independent - it only depend on the leakage current.