  In this paper, we partially complete the local Rankin-Selberg theory of Asai
$L$-functions and $\epsilon$-factors as introduced by Flicker and Kable. In
particular, we establish the relevant local functional equation at Archimedean
places and prove the equality between Rankin-Selberg's and Langlands-Shahidi's
$\epsilon$-factors at every place. Our proofs work uniformly for any
characteristic zero local field and use as only input the global functional
equation and a globalization result for a dense subset of tempered
representations that we infer from work of Finis-Lapid-M\"uller. These results
are used in another paper by the author to establish an explicit Plancherel
decomposition for $\mathrm{GL}_n(F)\backslash \mathrm{GL}_n(E)$, $E/F$ a
quadratic extension of local fields, with applications to the Ichino-Ikeda and
formal degree conjecture for unitary groups.
