  This paper is devoted to give several characterizations on a more general
level for the boundedness of $\tau$-Wigner distributions acting from weighted
modulation spaces to weighted modulation and Wiener amalgam spaces. As
applications, sharp exponents are obtained for the boundedness of $\tau$-Wigner
distributions on modulation spaces with power weights. We also recapture the
main theorems of Wigner distribution obtained in
\cite{CorderoNicola2018IMRNI,Cordero2020a}. As consequences, the
characterizations of the boundedness on weighted modulation spaces of several
types of pseudodifferential operators are established. In particular, we give
the sharp exponents for the boundedness of pseudodifferential operators with
symbols in Sj\"{o}strand's class and the corresponding Wiener amalgam spaces.
