= Solution
Construct the <tensor product of sheaves> by first forming the <presheaf> $U\mapsto\mathcal F(U)\otimes_{\mathcal O_X(U)}\mathcal G(U)$, with restriction maps induced by those of the two <sheaves of modules>, and then applying <sheafification>. The local <module> actions are compatible with restrictions and therefore give the sheaf an $\mathcal O_X$-module structure. Its <stalks> are
$$
(\mathcal F\otimes_{\mathcal O_X}\mathcal G)_x\cong\mathcal F_x\otimes_{\mathcal O_{X,x}}\mathcal G_x.
$$
Indeed, a finite collection of <germs of sheaf sections> can be represented on a common neighbourhood, and every finite tensor relation holds on a sufficiently small neighbourhood. Equivalently, this construction represents <bilinear maps> of <sheaves of modules> that are balanced over the <structure sheaf>. Sections of the resulting sheaf need not themselves be tensors of <global sections>: that is why the <sheafification> step matters.
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