Personally, I fizz fish through the mouth. For me, it's the simplest, most fool-proof method for on-the-water fizzing, because you can do it entirely in, or just above, the livewell, so it doesn't involve excessive handling of the fish, which can remove it's slime coat and further stress it.
There are those who say through the side is better, but I'm always afraid I'll miss the bladder that way, and kill the fish. Plus, having it on my thigh, or on the deck, removes the slime coat during the process.
As the weighmaster for a tournament trail, I saw a lot of fish that weren't fizzed because anglers were afraid they would kill the fish, so they let them float on their sides all day, which means they will suffer delayed mortality. They die after being released anyway.
I've seen the weights, and, no doubt, they work, but, especially in a tournament, just keeping a bass upright in the livewell doesn't relieve the pressure on it's internal organs from an over inflated swim bladder, which is fatal if left untreated.
Here's an illustration of where to insert the needle. Remember, you have to go through two membranes... the back of the throat, and then the air bladder itself.

- needle_placement.jpg (44.63 KiB) Viewed 3794 times
I recommend doing it in the livewell, underwater, so you can see the bubbles escape. I do it until the bubble stream starts to slow, but I don't completely deflate the air bladder the first time. Check on the fish after five minutes to see if it's able to swim okay, and needle it again if it's still floating.
Basically, the swim bladder is directly behind, and just above, the sphincter opening to the stomach. Stay just below the level of the two crushers, and just above the sphincter opening, and you'll do fine.
Many more fish die from not being fizzed than from mouth fizzing, if done right.