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Most black hole models don't have an "other side", they just have singularities that the things that fall into the holes get crushed into.
They aren't really holes at all, despite the name and that horrendous bowling-ball-ball-on-a-rubber-sheet analogy. It would be like if all the planets suddenly started falling into the sun. We wouldn't ask what's on the other side of the sun, we'd just say that all the planets end up in the sun, and then stay there.

There are, however, models sin which two black holes can connect to one another, forming what is called a "wormhole". To be strict regarding interpretation, if the black holes are far enough apart then one can't really designate which end is in the future or whatnot; to some people, one end is further along in time than the other, and to other people it will be the other way around, and neither view is more or less correct than the other. Of course, to the people traversing the wormhole, both ends occupy the same time period.
Notes on a few things:
1: these wormholes cannot be crossed by anything with positive energy, i.e. anything that exists. The very presence of energy nearby, whether it's kinetic energy or the energy inherent in matter, will collapse the wormhole tunnel on both ends.
2: Wormholes only link the singularities at the centers of the black holes. Even if you do get through a wormhole, you're still inside a black hole.

Actually, Hawking recanted the idea that information is lost in black holes. Now his theory is that the information that falls into black holes comes back out in the Hawking radiation from black holes, thus staying within this universe.

The idea is that while matter can't normally escape from black holes, there is a mechanism for energy to be taken from the black hole.
Virtual particle/antiparticle pairs appear near the event horizon; temporarily violating the law of conservation of energy; one of the pair falls into the hole while the other flies off, escaping the hole's gravity. Because the law of conservation of energy can only be violated for short times, normally this would require the pair to come together and annihilate, but since one got sucked into the black hole the free one can't be annihilated and becomes just like any other particle. Hence the energy instead gets taken from the black hole. The particles that arise in this fashion are called "Hawking Radiation".
The theory regarding black hole information now states that the information that goes into the black hole, i.e. the distribution of different kinds of particles that fall into the black hole, gets represented as the distributions of these particle/antiparticle pairs on the event horizon, and hence the Hawking Radiation carries off the information about the stuff that falls into the black hole.

Again, all of this happens within this universe, with no involvement of other universes. After all, if the information ended up in a different universe then we'd still have problems with conservation of information.