You say it's not multiplexed or a matrix, but judging from your descriptions, I'm not convinced they're simply connected to a common. In any case, if there are 'resistors' in this 'common' line, you probably can't use transistors simply as open collector switches, with their emitters common to gnd.....
Anyway, to simulate the connection of the two wires 'joining together' from each switch you described earlier, just use the 4066 CMOS switch I referred to above. Connect the two wires from each switch to the in/out terminlals of the switches within the 4066, and drive the control terminals from your PIC outputs.
Power the chip from the same power supply rail as the PIC.
That will eliminate the problems you seem to be creating, by trying to copy 'voltages' by using transistors.