The maximum independent set problem is a classic optimization problem that has also been studied quite intensively in the distributed setting. While the problem is hard to approximate in general, there are good approximation algorithms known for several sparse graph families. In this paper, we consider deterministic distributed CONGEST algorithms for the weighted version of the problem in trees and graphs of bounded arboricity. For trees, we prove that the task of deterministically computing a $(1-ε)$-approximate solution to the maximum weight independent set (MWIS) problem has a tight $Θ(\log^*(n) / ε)$ complexity. The lower bound already holds on unweighted oriented paths. On the upper bound side, we show that the bound can be achieved even in unrooted trees. For graphs $G=(V,E)$ of arboricity $β>1$, we give two algorithms. If the sum of all node weights is $w(V)$, we show that for any $ε>0$, an independent set of weight at least $(1-ε)\cdot \frac{w(V)}{4β}$ can be computed in $O(\log^2(β/ε)/ε+ \log^* n)$ rounds. This result is obtained by a direct application of the local rounding framework of Faour, Ghaffari, Grunau, Kuhn, and Rozhoň [SODA '23]. We further show that for any $ε>0$, an independent set of weight at least $(1-ε)\cdot\frac{w(V)}{2β+1}$ can be computed in $O(\log^3(β)\cdot\log(1/ε)/ε^2 \cdot\log n)$ rounds. This improves on a recent result of Gil [OPODIS '23], who showed that a $1/\lfloor(2+ε)β\rfloor$-approximation to the MWIS problem can be computed in $O(β\cdot\log n)$ rounds. As an intermediate step, we design an algorithm to compute an independent set of total weight at least $(1-ε)\cdot\sum_{v\in V}\frac{w(v)}{deg(v)+1}$ in time $O(\log^3(Δ)\cdot\log(1/ε)/ε+ \log^* n)$, where $Δ$ is the maximum degree of the graph.