Almost everyone does use two hands at some point (including me). But I am 90% sure that as you progress, you will switch to one handed again. You just don’t need much force from your arms when your technique is good.
Interesting question and answers.
I use my dominant hand (right) to hold the bars or saddle most often. It feels the most natural. I ride a 29er and 36er. I’ve only been riding for 10 months though so maybe my riding will change.
I’m currently working on riding with both hands on the bars / saddle.
— Realizes he’s been doind it the wrong way round for almost 40 years — Maybe that’s why I never got very good at Trials? Nah, I think the real reason is still that I haven’t spent enough time on it.
But it makes sense for the foot opposite the saddle-holding hand to be in front. This makes a better-structured “triangle” of support. I did always jump to the right, which seems obvious enough with the free hand being on the right.
I use my free hand (arm) to counteract “wobble” on uphills. I swing my right arm back as I push down on the right pedal, and bring it back as I push down on the left. This helps keep the power directed in a somewhat straighter line than otherwise.
There has been an attempt to name the two stances: Hopping with the foot forward on the same side as the hand on the saddle is asymmetric, and with the foot forward opposite side as the hand on the saddle is symmetric. (I hope I got it the right way around, but that naming is barely used anyway).
You will find both with good trials riders. Mark Fabian has the right hand on his saddle, and right foot forward (and hops to the left). Mike Taylor uses the same stance. Tim Desmet has the left foot forward, and the right hand on the saddle. Aidan Teleki actually learned to ride with either foot forward (while keeping the same hand on the saddle). All of these guys hop well over 110 cm. I forgot what Aidans reason was exactly for switching, but I believe one way is supposed to be higher over a bar, and the other onto an obstacle.
I’ve seen this sort of claim a few times - that practicing something like mounting on your “bad” side will help with the “good” side - but I just don’t believe it.
With muscle memory skills like this, you get better by doing something correctly, over and over. You won’t get better by doing something poorly and backwards. Nobody coaches e.g. right-handed tennis players to work on their serve by practicing serving left-handed. If skill transferred so easily, all that practice on your good side would have made you pretty good already on your other side, but it doesn’t work that way.
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There are some skills like mounting and hopping that I practice on both sides because they’re useful for muni, which is pretty much all I ride. My earlier comment about not bothering to learn both sides was just about which hand I use.
In boxing it is called Southpaw.
If I understand correctly, you are arguing that practicing on your good side doesn’t make you better on your bad side, so why should practicing on your bad side make you better on your good side?
My answer is this: Our bad side is also referred to as our weak side. Whereas we rely on strength for our good/strong side, success on the weak side is contingent more on improved technique and efficiency of motion. Once we learn that on our weak side, we can apply it to our good/strong side. However, the strong side cannot teach the weak side to be stronger.
Regarding your comment “You won’t get better by doing something poorly and backwards.” Applying that logic to practically every skill I’ve learned on the unicycle, I might question how I ever learned to ride in the first place, considering how awful and sketchy my technique has been on new skills.
Glad to hear you’re into muni. We have awesome riding trails here in SoCal.
“If skill transferred so easily, all that practice on your good side would have made you pretty good already on your other side, but it doesn’t work that way.”
This is the problem. You concentrated on one side to begin with and learned asymmetrically.
Skills don’t transfer, they’re developed.
Some sports and activities support asymmetry. Ping pong, tennis, shooting…
Some sports and activities support symmetry. Biking, paddling, climbing, etc…
I think muni is a contender for symmetry. (why wouldn’t you want to be able to handle anything that’s in front of you without stopping?)
I think trials is a contender for asymmetry. (quick learning and advancement is priority)
Finnspin is right about SIF hopping with only one hand -it really is better- but you are right about practicing on both sides, at least for this particular skill. Though I never have the patience to practice hopping with a reversed stance, and couldn’t even get up one step if I did, on a few occasions I have practiced hopping up steps to the left instead of to the right, and it did make me a lot smoother. I never went up 17 steps without a corrective hop, except one set of steps that is very gentle and smells like pee. For normal steps, I think my record is five without a correction, and to do that, I have to have done some fairly serious stair hopping in the last few days.
I’m not sure what your basing “being right” on, but I can sidehop 95cm over a bar, and I have never seriously practiced using my weak hand… 
I’m not saying you should not bother with practicing your weak side, and can’t care less about the will it help you with your strong side debate (just try it out, and if it helps you, that’s cool, if it doesn’t, probably didn’t hurt you to exercise the muscles on your opposite site). But what I can definitely say is that it’s certainly not necessary to progress on your strong side.
Practicing on my weak side, in the rare event that I bother to do it, seems to make me better at going up the steps without pausing to make corrective hops, but you may be right, I haven’t tested this finding with enough scientific rigor to say for sure whether the same amount of time practicing only on one side would not have helped just as much. For hopping down the stairs, I can easily do one hop per step, even if I am a bit out of practice, though only if I hop to the right.
I have never seen anyone, except on You Tube, who hopped higher than 30cm, which is not very high- even I can do it! High jumping is on my list of skills where I have tons of room for improvement. My tire is a 20 x 2.4, so not quite a proper trials tire, but if I learned to tuck, hopefully I could jump pretty high anyway.
All dominance on me is on the right hand side. Right is the first foot on the backpedal and i hold my ride hand on the seat, while flailing with left. On long rides i swtch around with my hands and often try to ride with both hands on the seat, which took me a while to learn because of having to balance with my hips instead of my arms.
Hopping i can only do with my right hand just like mounting with right. Mounting with left feels very strange. When hopping i find it easier to hop clockwise than anti-clockwise, but i do try both ways.
Its funny to see that there is a big variance between people here.