Not sure about the earlier T/Os, nor which one the 1st Div was using at the time of the photo, but the Jan '18 version for a rifle co shows the Thrower in the Hand Bomber Section is authorized both a rifle and a pistol. These guys are shown as PFCs, but that doesn't seem to be much of a reach.
No other privates in the division got both (by the book anyway).
With almost 12,000 pistols in this WWI division, it's not a big shock to see them show up.
Rifleman is a misunderstood term. In this WWI organization, about 6 of 7 rifles are in the hands of other than Riflemen. That's because only about 10% of this division was Riflemen.