  We use cosmological hydrodynamic simulations to consistently compare the
assembly of dwarf galaxies in both $\Lambda$ dominated, Cold (CDM) and
Self--Interacting (SIDM) dark matter models. The SIDM model adopts a constant
cross section of 2 $cm^{2}/g$, a relatively large value to maximize its
effects. These are the first SIDM simulations that are combined with a
description of stellar feedback that naturally drives potential fluctuations
able to create dark matter cores. Remarkably, SIDM fails to significantly lower
the central dark matter density at halo peak velocities V$_{max}$ $<$ 30 Km/s.
This is due to the fact that the central regions of very low--mass field halos
have relatively low central velocity dispersion and densities, leading to time
scales for SIDM collisions greater than a Hubble time. CDM halos with V$_{max}$
$<$ 30 km/s have inefficient star formation, and hence weak supernova feedback.
At a fixed 2 cm2/g SIDM cross section, the DM content of very low mass CDM and
SIDM halos differs by no more than a factor of two within 100-200pc. At larger
halo masses ($\sim$ 10$^{10}$ solar masses), the introduction of baryonic
processes creates field dwarf galaxies with dark matter cores and central
DM$+$baryon distributions that are effectively indistinguishable between CDM
and SIDM. Both models are in broad agreement with observed Local Group field
galaxies across the range of masses explored. To significantly differentiate
SIDM from CDM at the scale of faint dwarf galaxies, a velocity dependent cross
section that rapidly increases to values larger than 2 $cm^{2}/g$ for halos
with V$_{max}$ < 25-30 Km/s needs to be introduced.
