A theory with N=2 supersymmetry, where particles have two superpartners, has mirror symmetry built in. Nir Polonsky wrote some papers about an N=2 extension of the standard model (e.g.). The main problem for such a theory are the chiral Yukawa interactions between fermions and the Higgs field, which give fermions their mass in the SM. The mirror symmetry of an N=2 theory forbids direct interactions of this form, so they have to arise indirectly, after the breaking of supersymmetry, and this brings extra effects ("oblique corrections") which aren't observed.
Note that in these N=2 theories, a particle in the visible sector has a visible-sector superpartner, a mirror partner in the mirror sector, and a mirror superpartner in the mirror sector. In other words, one of the supersymmetry transformations acts within a sector, the other acts between them.
The intent of the question seems to be, could ordinary N=1 supersymmetry involve a mirror structure. But this would imply something resembling N=2 susy. Suppose we start with a boson B_visible in the visible sector, which has a superpartner F_mirror in the mirror sector. By mirror symmetry, there will also be B_mirror in the mirror sector and F_visible in the visible sector, which are also superpartners. But now we can ask, what's the relation between B_visible and F_visible, and between B_mirror and F_mirror? If these were also supersymmetric pairings, then we would have N=2 supersymmetry. If they aren't supersymmetric pairings, we have at least found that the combination of mirror symmetry and a supersymmetry across the mirror, means that there must be something like a supersymmetry (e.g. a matching up of boson and fermion degrees of freedom) within each sector.
So if the SM is contained in the visible sector, and we're following this theoretical path, we have a choice. We can have susy or a susy-like doubling within the visible sector, as in Polonsky's models, where the visible sector is the usual MSSM. Or, more exotically, we can look for susy or a susy-like relation already within the SM, or within some small extension of the SM.
A few ideas are coming to mind here. First, Stephen Adler recently proposed a susy-like GUT in which there is a matching of boson and fermion degrees of freedom. Second, just before the 1984 superstring revolution, there were attempts to get the SM from the Nicolai-Warner critical point of N=8 supergravity, which has N=2 supersymmetry, and Hermann Nicolai himself, at least, still likes to think that this has a chance of coming true. Third, Christopher Hill argues that there is a susy-like relation between top quark and Higgs boson in a particular limit. Fourth, Alejandro Rivero has found a susy-like mapping within the SM when certain composite (meson and diquark) degrees of freedom are included, which he calls the "sBootstrap".
My point is that if you took exotic wannabe supersymmetries like these seriously, you might be able to find an "N=2"-like structure lurking inside "SM + mirror SM" as studied by Foot and Volkas.