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Hello fellow 99 tahoe owner. I have the exact same problem. I'm in the process of working out what is specifically wrong. I have read hundreds of pages regarding this issue and have a good plan of action on this.
There are usually 2 common problems regarding this. Especially since you narrowed down the problems by stating the passenger seat is still working.
1) It could be the relay that is under each seat. This is the theory that I'm currently thinking may be my problem. This is better than having problem #2. You can purchase the relay (OEM for about $100) or I found a generic replacement that many people have used that say it works great for ~$5
The part is here:
https://www.surpluscenter.com/item.asp?UID=2009073013315492&catname=&qty=1&item=11-2443
I *think* I found mine up under the driver's seat right under the front lip. It looks like the other relays that are in your fuse box under the hood. You can test if this is the problem by taking the relay from the passenger side and swap both relays between the seats. If it was the relay, your driver seat will work and your passenger seat will not.
2) Problem #2 is the heating element. This is much more intensive to replace only because it requires you remove the leather from your seat completely and have it reupholstered which could add quite a bit of labor cost. I have seen elements stating they are made for the tahoe for $100-$150. If you were handy with leather you could do that for relatively cheap but taking it in could cost.
I'm hoping to be able to dig out the relays today and try that swap out and see if that helps. Those relays are quite commonly going bad. They are a bit tricky to get to under the seat and several people have mentioned taking their seat off the mount and tipping it back to get to it.
---------- Post added at 03:26 PM ---------- Previous post was at 03:19 PM ----------
Oh one more thing! This is what makes me think my relay could be the problem.
Turn your car on and let it run. Then plug in the seatbelt on your passenger side (the seatbelt has to be plugged in for the seat heater to operate). Now get your ear under your seat and flip the seat heater switch between off and on. You should hear the relay switch click. Do this several times so that you are sure of what you hear. It's pretty easy to hear.
Now goto your drivers side and do the same. See if you can hear the click. I could not on mine. That click between off and on is good. Check it out on yours.