I've actually done a good bit of testing on sparring in recent versions. I also - ironically enough - just recently responded in another thread on this same subject, so I'm going to be cheaty scrub and just copy paste over most of what I said there.
1. Sparring only occurs if you have two dwarves in the same squad, with orders to train, whose skills are relatively close, and who can path to each other. If you pair up people that have a wider difference in their combat skills, you'll end up with one of them organizing a combat-related skill demonstration; not awful, but not as amazing as sparring is.
2. Because of what I said in #1, squads of two dwarves each might seem like the best - but my testing has shown that you'd be wrong. In truth, the ideal squad size is three dwarves, where the squad is issued training orders that require only two positions. This is because you almost always have one of your two dwarves doing some that is not training - eating, boozing, sleeping, socializing, experiencing !!Parties!!, you get the idea. Giving yourself three dwarves with only two set to train allows one of the three-man squad to go have Urist-time while your remaining two can actually get some shit done together.
3. Make sure that your dwarves train in the same place. This I HAVEN'T done testing on, but I always designate only one place for my dwarves to train - apart from marksdwarves - and I seem to get a lot of sparring out of them. It wouldn't surprise me if this has some kind of impact on sparring frequency.
4. Especially when starting out, do not over armor your dwarves! While a set of light armor might not help as much against larger opponents, remember that in our current version, armor penalizes not only their movement speed, but how often they can act. The more weighed down your new recruits are, the few times they're going to take swings at each other, and the slower they're going to level up. Your 'light' armor set should look something like this (make sure it is set to replace clothing!):
shirt
metal chain mail
leather armor
metal helm
hood
mittens
metal gauntlets
trousers
metal leggings
sock
sock
metal low/high boot
metal low/high boot
wooden shield
weapon of your choice
(Notes: the double sock/low boot helps with the bug you may have seen where only one foot gets protection. This solves that problem. I'm not entirely sure why. Also, make the clothing items out of leather, if possible, as they offer a slight blunt resistance advantage over cloth, which is what they're really for. Also, this getup gives your dwarves basic modesty with a shirt and trousers, so if your armorers are behind you don't end up with tantruming, naked dwarven warriors wielding silver maces.)
Even with this, you might still notice novice armor users moving a bit slow, but that's the price you pay for not needlessly slaughtering your new recruits.
5. Disable all labors for dwarven warriors (except for any on-purpose cross training you might be doing). I've noticed that, given the barest chance a dwarf, even a dwarf assigned on duty, will still occasionally go off to do a non-military job, ESPECIALLY HAULING. While I'm on the subject, make sure that the squad is active and has a designated training area. I've been playing for years, and I've still made this mistake.
I do all the above, and I have adept and higher-level weapons users after a handful of years following conscription, largely due to sparring.