Its going to be all but impossible to answer that question, unfortunately. Slingers were most  often auxiliaries i.e. mercenaries and werent part of the formal legion structure. They could be poor quality local bodies recruited cheaply or more expensive reputable companies like the Balearic slingers. Unless it is directly mentioned that such-and-such an army had 'x' hundred slingers then you are into the realms of guesswork.

I also think it is quite impossible to quantify. There must be big differences in the levies of troops in the different wars which took place in the 3rd century (esp. against Pyrrhus, and then against the Carthaginians in the 1st and even more in the 2nd Punic War), and the possibilities for Rome to recruit slingers at all.  
 
As the Balearic Islands were in Carthaginian hands until the end of the century, Rome probably had no access to the mercenaries there, but maybe to Rhodian or Cretan slingers, apart from the slinging auxilia they could raise in Italy itself.
 
I would say that they generally used a smaller number of slingers than in later centuries, but that is just a guess.

Powerful enough that specific medical instructions were written for the removal of sling projectiles.

I'd bet a good 300,000 shepherds over the imperial age of Rome were lucky enough to set out on the trip of a lifetime.

I once asked a local roman reenactment group this question, and they said that it was not uncommon for there to be as many slingers as archers. Mainly for strategic reasons. While the enemy raised thier shileds to block the incoming arrows the slingers would aim for their exposed chests or heads.

In Parziwal of Wolfram von Eschenbach, Slinger are mentioned as a special weapon type.
 
Ash brook writes: her number was big and they were feared on all battlefields.
 
 
In an other work of ash brook he mentions like the woman of the count Willehalm with her virgins by this fling from stones, to many moors then death brought.
 
So the Sling not only a weapon of farmers.
 
The book is called will stalk.
 
Full of suspense also, because ash brook in in the book battles themselves described joined in the struggle.
 
It is an eyewitness report

Quote from kuggur slingdog on Jul 29^th, 2010, 10:14am:

 
You have to remember that you can see an arrow in flight and not a sling bullet. Part of an arrow vollies effectivness is the terror it induces in the recieving side.
 
http://www.youtube.com/watch?v=mKFNmtrN3K8
http://www.youtube.com/watch?v=kWyFWFJ0Ywk