  We consider the problem of $T$-Private Information Retrieval with private
side information (TPIR-PSI). In this problem, $N$ replicated databases store
$K$ independent messages, and a user, equipped with a local cache that holds
$M$ messages as side information, wishes to retrieve one of the other $K-M$
messages. The desired message index and the side information must remain
jointly private even if any $T$ of the $N$ databases collude. We show that the
capacity of TPIR-PSI is
$\left(1+\frac{T}{N}+\cdots+\left(\frac{T}{N}\right)^{K-M-1}\right)^{-1}$. As a
special case obtained by setting $T=1$, this result settles the capacity of
PIR-PSI, an open problem previously noted by Kadhe et al. We also consider the
problem of symmetric-TPIR with private side information (STPIR-PSI), where the
answers from all $N$ databases reveal no information about any other message
besides the desired message. We show that the capacity of STPIR-PSI is
$1-\frac{T}{N}$ if the databases have access to common randomness (not
available to the user) that is independent of the messages, in an amount that
is at least $\frac{T}{N-T}$ bits per desired message bit. Otherwise, the
capacity of STPIR-PSI is zero.
