Question: Does a newly drafted recruit improve in skill faster if I only have him spar with champions?
Generally no. Nor does the champion.
I have a lot of experience with this with Morul. He and another soldier would head off to spar, Morul would get one hit in and the other guy would have to break off sparring to heal. Now, if we could set up sparring to be one-sided, where the lesser skilled solider could just wail on the higher skilled guy, that'd be ideal. The whole fortress could beat on Morul all year long and he'd never get hurt, so they'd skill very steadily on weapon, but not at all on shield/armor. Morul would skill shield/armor very steadily, just not weapon. If your draftee is already Ultra-Mighty due to other skills, it might work out alright - but what you're trying to do is get the offensive skill of one to balance out the defensive skill of the other. When offense of one > defense of the other, things are inefficient.
To help balance things, put your sparrers in the best armor you can manage. The less they get hurt, the longer they spar. Give them the weakest weapons you can manage - no quality silver or wood is the best you can do. If one dwarf is much stronger than the other, that's when problems develop from what I've seen.
warlord is right that more = faster. Make sure that there's always a sparring partner and things will go faster. If your dwarf is skilling shield much faster than armor, just unequip the shield and that'll balance out.