In such a model, you will end up with an E8xE8 gauge field on a space-time which is (d=4 Minkowski space-time) x (some Calabi-Yau).
Apparently the key is that you choose a vector bundle on the CY which can then be embedded into E8xE8, and this defines how the E8xE8 gauge field is attached to the space-time. SU(3) bundles are just the first possibility that was considered.
These days they might consider a bundle of the form SU(4) + U(1), with the SU(4) embedding into the first E8 leaving an SO(10) GUT, and the U(1) embedding into the second E8 leaving an E7 x U(1) GUT. (That example comes from
here.)
At this point, you have a 4-dimensional field theory with a sector consisting of SO(10) GUT with N=1 supersymmetry, and another sector consisting of an E7 x SU(1) GUT with N=1 supersymmetry, and the only interaction between the two sectors is through gravitation.
The matter of the visible universe comes from the SO(10) GUT. Particles from the E7 GUT will be part of the dark matter, perhaps along with superparticles from the SO(10) sector. So you would end up with a multi-component model of dark matter, e.g. as a mix of neutralinos from the visible sector and E7 glueballs from the hidden sector.
The idea that the other E8 would contribute to dark matter has been around since the start of heterotic phenomenology, but oddly, I haven't found any detailed cosmological calculations for this sort of multi-component model. I think people have been too preoccupied with getting the visible E8 to work, to care very much about the details of what happens to the hidden E8.
Another idea which has been discussed, is that supersymmetry is broken in the hidden sector - e.g. in this case by condensation of E7 gauginos - and then supersymmetry-breaking is transmitted to the visible sector by the gravitational interaction (this is called "gravitationally mediated supersymmetry breaking"). That would give the other E8 something to do, apart from just contributing to dark matter.
