If both turn signal sides don't work, then it may be the flasher but its a good idea to perform the following test anyway to make sure.
Do both the parking lights (low intensity) and DRLs (high intensity) work?.
1.) Turn on the hazards and walk around the car
2.) Look for any bulbs that aren't blinking (that are supposed to blink) or are blinking erratically
3.) When you find the problem bulb, turn off the hazards
4.) Inspect the problem bulb and its socket. Either the bulb blew or the socket has corroded. There are two filaments in most of the signal bulbs. Its possible that the bright filament (signal) broke and the dim filament (parking) didn't.
5.) Replace the bulb or repair/replace the socket (which ever was the problem). Usually its just a blown bulb.
6.) Use dielectric grease on the bulb base and socket to help keep water out of the connection.
If everything works properly with the hazards, but neither side works with the signals, then the signals flasher may need to be replaced.
The 4th-gen f-body has two flasher modules. One for the left/right signal and one for the hazards. The same flasher operates both the left and right signals. That is why it cannot be the problem if one side works and the other doesn't.
Before you start tearing into the dash and under it, be sure to confirm you have no power at the actual sockets.
To test the turn signal switch,
Gain access to the bottom of the steering column. Follow the wires coming from under the steering column until you find the harness.
There you will find the inputs and outputs of the switch. Below is the color code of the wires you will need to test.
Purple - Input power to the switch.
Outputs from the switch:
Light Blue - Left Front blinker
Dark Blue - Right Front blinker
Yellow - Left Rear blinker
Green - Right Rear blinker
Take a test light get a good ground, with the ign in the on position, engage the turn signal side you will test.
First test for power hitting the switch purple wire.
Then test outputs for each blinker. You should have power on all of the above to make sure the switch is good.
The only real difference between the front signals and the rear is that the front gets routed through the Daytime Running Light module. It's possible that there is some problem with the DRL module especially if someone tried to disable the DRLs.. You can test these possibilities at the DRL module. Remove the radio and you will find the DRL module mounted to the defroster duct behind the radio to the right of the opening. It is a black box about the size of a pack of cigarettes with two 8-pin connectors (one black and one blue). You're going to want to use a test light or multimeter with a sharp point so that you can check outputs with the connector still attached to the module. Turn on the ignition and the left turn signal. Check that you get power at the light blue with white wire in the black connector. If so then the turn signal switch is working properly. Next, test for power output at the light blue wire (no white stripe) - also in the black connector. If not then the DRL module is faulty and should be replaced. You can confirm this by unplugging the black connector and shorting the light blue with white wire to the light blue wire to see if this makes the turn signals work (in other words, bypassing the DRL module). Note that in both cases this power would normally be flashing but yours don't flash so we just want to check continuity.
If you don't get power on the light blue with white wire at the DRL module then you should suspect the turn signal switch. You can test this at connector C216 which is an 11-pin black single-row connector at the base of the steering column. With the ignition and left turn signal on, check that you have power in the purple wire (this is the power input to the switch from the flasher). Then check for power in the light blue with white wire (this is the left signal output from the switch). If you have power going in but not coming out then you can safely assume that the switch is faulty.
It's probably not the grounds. The two sides are not directly related to each other (i.e. they are completely separate circuits from the DRL module forward and have separate grounds)