Do you have schematics?
My first guess would be the you have your logic inverted for the low side. The COM pins connect to ground. Which means to turn on that segment in COM1/PIN1 you would play a 0 on its pin and a 1 on all the others (COM2/PIN2, COM3/PIN3, COM4/PIN4).
Turning on multiple segments may lead to too much current sinking through the PIC16 pin connected to COM1, COM2, COM3, and COM4. So you will probably need a sink transistor for each of these.
To turn on 3A you would put a 0 on COM1/PIN1 and 1 on PIN7, COM2/PIN2, COM3/PIN3 and COM4/PIN4. (0 on PIN5, PIN6, PIN8-14 to just light 3A)
This display is multiplexed so you will need to scan through PIN1-PIN4 (Only one of them can be 0 at any point), each time you will need to change PIN5-PIN14.
I hope this helps.