@CharlesRezk As Dylan writes, $X\colon C^{\mathrm{op}}\to \mathcal{S}$ preserving products should imply that $C_{/X}$ admits finite coproducts, and is thus filtered. For the converse, denote by $F: \mathrm{Fun}(C,S) \to S$ the Yoneda extension of $X$, then $F$ can be written as a sifted colimit of representable functors, all of which preserve all limits, and thus itself preserves finite colimits. We then have $X = F \circ j$, where $j$ is the Yoneda embedding, which preserves all limits.

(More precisely, $F$ can be written as a colimit of corepresentable functors on $C^{\mathrm{op}}$.)

@AdrianClough So even if we drop the requirement that $C$ has finite coproducts, it is still true that $(C/X)$ is sifted iff $F\colon \mathrm{Fun}(C,\mathcal{S})\to \mathcal{S}$ preserves finite products?

@CharlesRezk As far as I know, the identification of $P_\Sigma(C)$ (the ∞-cat freely generated under sifted colimits) with those presheaves whose lke to $P(C)$ respects finite products is still an open question (it's even known to be false in the 1-categorical case for presheaves of sets). This makes me afraid that the question might be nontrivial to work out

Is it true that $(C/X)$ is sifted iff $X\in P_\Sigma(C)$?

(@AdrianClough "admits finite coproducts, and is thus filtered" should say "admits finite coproducts, and is thus sifted")