This was originally a PM, but I think discussion of this can help people, so I've moved it here, into the public, as I don't think @cipher will mind.
It's a good question.
For me, and this will change a little (only a little) depending on your pattern, but the ideal position for the right thumb tends to be just forward of center. So, more the right-hand image than the left-hand image here:
You can see how the right thumb moves up the aft side of the line on the left, and the fore side of the line on the right. Why is this typically the better position? Because it helps support the club a bit more at the top of your backswing. With the thumb on the aft side of the shaft, depending on the type of player you are, you'll either push against it causing a steepening of the shaft, or you'll want to feel pressure against it and support the club more and you'll lay the shaft off. I will stress again that everyone's a bit different here, and I'm only speaking to a generality, or what would be considered a "neutral" or "average" right thumb.
It's interesting to note the left thumb as well. In the left image, you can see that it's not covered by the heel pad. In fact, in the left image, it's almost a putter grip with both thumbs going down the middle of the shaft (blue arrow). I'd prefer to see that a bit more on the aft side of the shaft, so it fits in the "pocket" of space between your thumbs.
Here are a few pictures of that:
The "thumb track" or the valley formed when you cup your hand is shown in the first two pictures.
Essentially, then, from a right-hander's perspective looking down at their grip:
Your left thumb will be slightly /.
Your right thumb will be slightly \.
The shaft will be splitting the middle: |.
Again, that's the generality. Can you play golf differently? Sure. I've broken par keeping my right thumb literally floating off the club entirely on anything outside of 50 yards (it was injured). These are just the generalities. The primary functions of the thumbs are to support the club at the top of the backswing, and to allow the golfer to sense force and direct force in the proper direction.