  We study the effect of decoherence on dynamical phase diffusion in the
two-site Bose-Hubbard model. Starting with an odd parity excited coherent
state, the initial loss of single particle coherence varies from small bound
oscillations in the Rabi regime, through hyperbolic depletion in the Josephson
regime, to a Gaussian decay in the Fock regime. The inclusion of local-site
noise, measuring the relative number difference between the modes, is shown to
enhance phase-diffusion. In comparison, site-indiscriminate noise measuring the
population imbalance between the two quasi-momentum modes, slows down the loss
of single-particle coherence. Decoherence thus either enhances or suppresses
phase-diffusion, depending on the details of system-bath coupling and the
overlap of decoherence pointer states with collisional-entanglement pointer
states. The deceleration of phase-diffusion due to the coupling with the
environment may be viewed as a many-body quantum-Zeno effect. The extended
effective decay times in the presence of projective measurement, are further
enhanced with increasing number of particles $N$, by a bosonic factor of
$\sqrt{N}$ in the Fock regime and $N/\log{N}$ in the Josephson regime.
