  We consider the problem of noisy private information retrieval (NPIR) from
$N$ non-communicating databases, each storing the same set of $M$ messages. In
this model, the answer strings are not returned through noiseless bit pipes,
but rather through \emph{noisy} memoryless channels. We aim at characterizing
the PIR capacity for this model as a function of the statistical information
measures of the noisy channels such as entropy and mutual information. We
derive a general upper bound for the retrieval rate in the form of a max-min
optimization. We use the achievable schemes for the PIR problem under
asymmetric traffic constraints and random coding arguments to derive a general
lower bound for the retrieval rate. The upper and lower bounds match for $M=2$
and $M=3$, for any $N$, and any noisy channel. The results imply that
separation between channel coding and retrieval is optimal except for adapting
the traffic ratio from the databases. We refer to this as \emph{almost
separation}. Next, we consider the private information retrieval problem from
multiple access channels (MAC-PIR). In MAC-PIR, the database responses reach
the user through a multiple access channel (MAC) that mixes the responses
together in a stochastic way. We show that for the additive MAC and the
conjunction/disjunction MAC, channel coding and retrieval scheme are
\emph{inseparable} unlike in NPIR. We show that the retrieval scheme depends on
the properties of the MAC, in particular on the linearity aspect. For both
cases, we provide schemes that achieve the full capacity without any loss due
to the privacy constraint, which implies that the user can exploit the nature
of the channel to improve privacy. Finally, we show that the full unconstrained
capacity is not always attainable by determining the capacity of the selection
channel.
