The GFCI works by comparing the current/amperage going out on the hot wire (usually black) with the current/amperage on the neutral/common (usually white). It makes this comparison 40 times each second. If the currents are unbalanced by 5 mA (+/-1) then it immediately trips and opens the contacts for both the hot and common (but leaves the ground connected). There is a MOSFET circuit involved to handle the math.
Because of the circuitry involved, it would take two completely separate GFCI units to provide protection to two circuits as you described. Other options would be:
1. Use a GFCI cordset as others suggested
2. Use a GFCI circuit breaker (likely the most costly option)
3. Use your standard GFCI and then use the 'load' terminals to supply two separate standard outlets. Everything connected to the load terminals receives the same GFCI protection. In fact, you could use one outlet and break the tabs as proposed above. Using this method, you would run each feed through the switch before you hit the outlet but after the GFCI.
Hope this helps.