The plural in the title hints that there are several complexities involved here.
The most traditional one is the computational complexity of the verifier; but, in the context of Interactive Proofs of Proximity (IPPs), the query complexity of the verifier and the communication complexity of the system should take the front seat.
Hence, we avoid limiting the computational complexity of the verifier, since this limitation blurs the role of the main complexity measures that should be of interest in the case of IPPs.

Specifically, we consider IPPs in which the verifier strategy is only restricted by its query complexity and its communication complexity.
While we do not limit the computational complexity of the verifier, we do care about the computational complexity of the properties for which we prove a query+communication lower bound.
Our results present a trade-off between the computational complexity of the properties for which query+communication lower bounds are asserted and complexity assumptions that condition these results.
Specifically, letting $q$ and $c$ denote the query and communication complexities of IPPs (and letting $n$ denote the input length), we essentially prove
\begin{enumerate}
\item An unconditional lower bound of $q(n)+c(n)=\Omega(n/\log n)$ for properties that are decidable in double-exponential time.
\item Assuming that exponential-time (resp., NP) contains a set that does not have (non-uniform) small-exponential circuit size, we obtain a $q(n)+c(n)=\Omega(n/\log n)$ lower bound for properties that are decidable in exponential-time (resp., in NP).
\item Assuming the existence of pseudorandom generators that fool small-exponential size circuits,
we obtain a lower bound of $q(n)+c(n)=\Omega({\sqrt{n/\log n}})$ for properties that are decidable in P.
\end{enumerate}
In contrast, recall that any property in uniform-NC has an IPP with $q(n)+c(n)={\widetilde O}({\sqrt n})$.
Hence, the almost linear lower bounds of the first two items are unlikely to be proven unconditionally for properties in NP.

In the foregoing text, $c(n)$ denotes the total number of bits exchanged between the two parties (in both directions).
One may wonder whether similar bounds hold when considering only the number of bits communicated by the prover, denoted $p(n)$.
We show that this is essentially the case, by proving that the verifier gains almost nothing by sending
messages of length exceeding $O(q(n)\cdot\log n)$.
Specifically, we show that if $\Pi$ has an $r(n)$-round IPP in which the prover sends at most $p(n)$ bits
and the verifier makes at most $q(n)$ queries, then $\Pi$ has an $r(n)$-round IPP of query complexity $O(q(n))$ and communication complexity $O(p(n)+r(n)\cdot q(n)\cdot\log n)$.