  We consider a Josephson contact mediated by 1D chiral modes on a surface of a
3D topological insulator with superimposed superconducting and magnetic layers.
The system represents an interferometer in which 1D chiral Majorana modes on
the boundaries of superconducting electrodes are linked by ballistic chiral
Dirac channels. We investigate the noise of the Josephson current as a function
of the dc phase bias and the Aharonov-Bohm flux. Starting from the scattering
formalism, a Majorana representation of the Keldysh generating action for
cumulants of the transmitted charge is found. At temperatures higher than the
Thouless energy $E_{\rm Th}$, we obtain the usual Johnson-Nyquist noise, $4G_0
k_{\rm B}T$, characteristic for a single-channel wire with $G_0 \equiv
e^2/(2\pi \hbar)$. At lower temperatures the behavior is much richer. In
particular, the equilibrium noise is strongly enhanced to a
temperature-independent value $\sim G_0 E_{\rm Th}$ if the Aharonov-Bohm and
superconducting phases are both close to $2\pi n$, which are points of emergent
degeneracy in the ground state of the junction. The equilibrium noise is
related to the Josephson junction's impedance via the fluctuation-dissipation
theorem. In a striking contrast to usual Josephson junctions (tunnel junctions
between two $s$-wave superconductors), the real part of the impedance does not
vanish, reflecting the gapless character of Majorana modes in the leads.
