This paper addresses the cornerstone family of \emph{local problems} in distributed computing, and investigates the curious gap between randomized and deterministic solutions under bandwidth restrictions. Our main contribution is in providing tools for derandomizing solutions to local problems, when the $n$ nodes can only send $O(\log n)$-bit messages in each round of communication. We combine bounded independence, which we show to be sufficient for some algorithms, with the method of conditional expectations and with additional machinery, to obtain the following results. Our techniques give a deterministic maximal independent set (MIS) algorithm in the CONGEST model, where the communication graph is identical to the input graph, in $O(D\log^2 n)$ rounds, where $D$ is the diameter of the graph. The best known running time in terms of $n$ alone is $2^{O(\sqrt{\log n})}$, which is super-polylogarithmic, and requires large messages. For the CONGEST model, the only known previous solution is a coloring-based $O(Δ+ \log^* n)$-round algorithm, where $Δ$ is the maximal degree in the graph. On the way to obtaining the above, we show that in the \emph{Congested Clique} model, which allows all-to-all communication, there is a deterministic MIS algorithm that runs in $O(\log Δ\log n)$ rounds.%, where $Δ$ is the maximum degree. When $Δ=O(n^{1/3})$, the bound improves to $O(\log Δ)$ and holds also for $(Δ+1)$-coloring. In addition, we deterministically construct a $(2k-1)$-spanner with $O(kn^{1+1/k}\log n)$ edges in $O(k \log n)$ rounds. For comparison, in the more stringent CONGEST model, the best deterministic algorithm for constructing a $(2k-1)$-spanner with $O(kn^{1+1/k})$ edges runs in $O(n^{1-1/k})$ rounds.