Hi prakhag - good question. Each school handles it differently, but typically you'd spend one semester in the foreign school (and usually, the home school receives one student from the foreign school in your place). Most schools only allow this during your second year of studies, and we frequently see people do it in the very last term of the MBA, since the majority of recruiting happens in the fall - most people would want to take advantage of that from their actual school, not miss out on it by being in the foreign school. We're not sure if the foreign school would even let you use their career services; their school would be working hard to find job opportunities for their graduates, and they might not want you, who isn't even officially their student, to take them.
That being said, you'd certainly be able to network and make connections with the students at the foreign school, and you'd be exposed to lots of local employers in Singapore. If you were resourceful (which obviously you are, given how you're thinking about this stuff!!) then you might be able to make this happen for yourself as you've envisioned it.
We would suggest that you contact the schools themselves (Cornell, in this case) and ask them about this. It's not what most people do but that doesn't mean that you couldn't do it.
Please report back on what you discover! We'd be curious to hear what the schools tell you.
EssaySnark