= Solution
The <degree-one generation condition for Proj> ensures that the opens $D_+(f)$ for $f\in S_1$ cover $X$. Put $B=S_f$ and $R=B_0$. Multiplication by $f^n$ is an $R$-linear bijection $R\to B_n$ for every integer $n$, because $f$ is an invertible degree-one element. The convention $S(n)_d=S_{n+d}$ therefore identifies $\widetilde{S(n)}$ on this chart with the free rank-one $R$-module $B_n$, generated by $f^n$. Hence \b[every twisting sheaf $\mathcal O_X(n)$ is invertible], including negative twists.
For the <tensor product of sheaves>, consider the natural map
$$
(M_f)_0\otimes_R(N_f)_0\longrightarrow(M_f\otimes_BN_f)_0.
$$
It is an isomorphism by <degree-one localization of a graded module>. Explicitly, a homogeneous tensor $m_a\otimes n_{-a}$ of total degree zero is represented on the left by $f^{-a}m_a\otimes f^a n_{-a}$. Multiplying a tensor factor by a homogeneous element of $B$ gives the same result after using the tensor relation, since that element is a degree-zero coefficient times a power of $f$. This defines the inverse. Since <localization commutes with tensor products>, the target is $((M\otimes_SN)_f)_0$. These natural chart maps agree on overlaps, so
$$
\boxed{\widetilde M\otimes_{\mathcal O_X}\widetilde N\cong\widetilde{M\otimes_SN}.}
$$
The degree-one hypothesis matters: on general <Proj constructions>, twisting sheaves need not be line bundles. The local trivializations are also recorded in https://stacks.math.columbia.edu/tag/01MM[Stacks Project, Section 27.10].
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