Hi Watchfixer;
You need a very thin very fine stone. I have a slip of ceramic that's about the grit of a moonstone. A thin piece of any metal with 2000 grit silicon carbide glued on would probably work as well, it isn't as though sharpening tweezers is done frequently enough to wear it out in any hurry.
Typically, you will remove such a minuscule amount of material that it seems almost a waste of time - you think "all that for a faint silver smear or two on a stone?". The point is that we're not grinding or hogging material here unless you have a pair that is so torn up that you're making a heavy blunt point pair from a #2 or the such.
The first thing to do is fix any bent tips. Sometimes a tip will be curled up.I put a bit of sheet steel between the tips and gently work the tip back down. There is almost always a bit of spring back, but the roughing in of the tip must be backed up to avoid creating a wavy end. Once the tip is close, you can gently rub the outer face of the tip against a polished or at least very smooth hard surface. Check often until the tips look good.
Next, flatten and level the inner surface where the tweezer grips. Gently hone the inner faces of the tweezers flat. Remember that the center of the joined end of the tweezers will usually be below the sharpening plane as the tips should meet at the tip first. Just a few gentle passes on each face, then check with light from behind the tweezers to see that the tips meet and black out first, then the contact area slowly grows as the tweezers are squeezed a bit more. It may only take three or four gently passes on each face to get things right. Remove as little as possible, if the face geometry changes, the feel and grip force of the tweezer with change.
You may be done at this point, or you may want to dress up and repoint the tweezers too. It can be helpful to hold a small bit of flat stock, wood, a coin, just a spacer of some kind in the tweezers so the tips just meet when you grip the tweezers, particularly with very fine tips. This keeps the change in your grip force from changing the tip form as you roll the tweezers in your hand. Then using a very fine stone or super fine lapping plate (even the silicon carbide on a plate if that's all you have) stroke the sides to shape, then the top and bottom so that the line where the tweezer tips meet is as close to invisible as possible. Having a new pair of similar tweezers or a large catalog picture at hand helps you decide just what should be rounded, and by how much.
It isn't needed often, but as soon as you notice that a pair of tweezers just doesn't feel quite right touching up the gripping surfaces can make a world of difference is how pleasant or irksome your day ends up

Quite often in the microelectronics world, we would gently close a pair of tweezers on a piece of 20 pound paper and pull the paper through the tips several times. While more of a cleaning operation, there is a small bit of abrasive action as well. It made a perceptible difference in grip.
Hope this is helpful,
Stan Stocker
Bentonville, Virginia