  Intrinsic noise is known to be ubiquitous in Josephson junctions. We
investigate a voltage biased superconducting tunnel junction including a very
small number of pinholes - transport channels possessing a transmission
coefficient close to unity. Although few of these pinholes contribute very
little to the conductance, they can dominate current fluctuations in the
low-voltage regime. We show that even fully transparent transport channels
between superconductors contribute to shot noise due to the uncertainty in the
number of Andreev cycles. We discuss shot noise enhancement by Multiple Andreev
Reflection in such a junction and investigate whether pinholes might contribute
as a microscopic mechanism of two-level current fluctuators. We discuss the
connection of these results to the junction resonators observed in Josephson
phase qubits.
