Can I split a 3 wire circuit into 2 separate 2 wire circuits?
As others have mentioned, this is a shared neutral configuration.  Technically, yes you can do it however I believe the current NEC only permits such circuits in a kitchen. The most common use of this is to have an outlet where both top and bottom plugs are on different circuits. 

You are also required to have the circuit breakers tied together so if one trips, both are shut off.  The most important part of this configuration is that the two hot conductors come from different legs of your service so that they are out-of-phase from each other.  The result of this is that they balance out the neutral conductor; if you used hot from the same phase you could double the allowed current over your neutral conductor.