According to the 5th edition of Military Small Arms of the 20th Century the first Mosin Nagant Sniper Rifles did not appear until 1937/38. These first units were fitted with the PU scope, later versions were upgraded to the PE scope. According to the information in this book these rifles were no different than standard issue Mosin Nagant. With the exception of the scope mount and bolt handle.
My own thought on this are.
Aside from these being no different than the off the shelf Mosin one could speculate that the these Snipers were probably cherry picked from the assembly line as they were tested. I just can't imagine sending a sniper out with a rifle that does not cycle well or isn't just a tad bit more accurate than the others. I base this on the fact that there were a lot of enemy solders at Stalingrad who didn't come home because of the shooting acumen of these rifles and their bearers.
Seeya
Mark