7 segment LED display not working when using common pin as cathode

Here is a possible mechanism for the fault. You say it works (dimly) without the transistors, but doesn't work with them. So it sounds like the fault is in the transistors. Is there some reason that the transistors might be letting some current through when they shouldn't?

Yes. You're using a PNP transistor. As you know, these transistors are on when the base voltage is lower than the emitter voltage. They are off when the base voltage is higher than or equal to the emitter voltage.

The problem with the shift register chip is that the outputs are always lower than the emitter voltage. I couldn't quite make out the part number of the chip you're using, but according to the datasheet for the 74HC595 (page 6), the outputs don't quite reach Vcc. If there is a tiny voltage difference, then you could find that a small amount of current is leaking out of the base of the PNP transistor. With a gain of about 100, you could find that there's enough CE current to give noticeable light output on the LEDs.

Something to try: Add a schottky diode between Vcc and the emitter. This should drop the collector voltage by a fraction of a volt, just enough to allow the shift register to fully turn off the transistor.

Actually this is a very common arrangement. Each digit has its anodes connected together, and the segments for all digits have their cathodes connected. The controller enables one of the common anodes, drives the segments, and waits, then turns off the cathodes and moves on to the next common anode. This arrangement and control method is called multiplexing, and the linked article will give you a lot of information. The part in question is a common anode display, but common cathode displays are just as common.

Multiplexing allows you to drive a large number of display elements with fewer wires than individual addressing, at the expense of a more complex control algorithm. You can go a step further by doing something called Charlieplexing which allows you to drive even more elements with less I/O if you can turn off the drivers (as opposed to just setting them high or low, which is the requirement for a multiplexed display controller).

Now that you know it's called a multiplexed display, you should have no trouble finding all kinds of example circuits or microcontroller code fragments to implement it.