  Johnson noise thermometry enables direct measurement of the electron
temperature, a valuable probe of many-body systems. Practical use of this
technique calls for non-equilibrium generalizations of the Johnson-Nyquist
theorem. For a hydrodynamic Corbino device, however, a na\"ive use of the
Shockley-Ramo theorem alongside the "Corbino paradox" leads to yet another
paradox: bulk velocity fluctuations cannot be measured by the contacts. In this
work, we resolve the unphysical "Corbino Shockley-Ramo paradox" by correctly
formulating the hydrodynamic Shockley-Ramo problem. This allows us to properly
formulate the problem of current noise in an hydrodynamic multi-terminal device
of arbitrary geometry, as well as validate a previously unjustified assumption
for rectangular geometry results. As an example, we compute the Johnson noise
in a hydrodynamic Corbino device, where we find a suppression of Johnson noise
with magnetic field. This unusual characteristic serves as a strong signature
of viscous hydrodynamic behavior.
