Potentiometer Circuit Design
Here's the simplest way to get what you asked for:-







simulate this circuit &ndash; Schematic created using CircuitLab

Each side of VR1 forms a voltage divider in combination with R2/R3 or R1/R4. When the pot is centered each side is 5K so the signals at Vout1 and Vout2 are attenuated by less than 10%. Thus each output is more than 90% of Vin, and (assuming the pot can be precisely centered) exactly the same as the other. 

As the pot is moved off center the resistance of one side decreases, causing more attenuation on that side. At full rotation that side is short, and then the series resistors divide the voltage by two. The other side now has less than 5% attenuation, ie. a slightly higher voltage than when the pot was centered, but that is still within your specification of being 90~100% of the input voltage.   

If this was an audio circuit (eg. balance control) it would sound like one side stays at constant volume while the other side gets quieter. However the attenuation control is very non-linear, being 'soft' in the center and getting more aggressive towards either end. 

Trying to make the attenuation control more linear is where you are likely to 'run into a wall'. You could use the pot to generate DC control voltages that can be shaped to suit whatever response you want, but then you need voltage controlled amplifiers or attenuators that can handle 1V signals with low distortion - which are not easy to make using only discrete components.