  We construct analogues of Rankin--Selberg integrals for Speh representations
of the general linear group over a $p$-adic field. The integrals are in terms
of the Shalika model and are expected to be the local counterparts of (suitably
regularized) global integrals involving square-integrable automorphic forms and
Eisenstein series on the general linear group over a global field. We relate
the local integrals to the classical ones studied by
Jacquet--Piatetski-Shapiro--Shalika. We also introduce a unitary structure for
Speh representation on the Shalika model, as well as various other models
including Zelevinsky's degenerate Whittaker model.
