I got a putting lesson today. I put Left hand low. My instructor says that generally, the left hand is dominant in left hand low strokes, and right hand is dominant in "conventional" strokes.
I told him that I still feel my right hand providing the feel for my stroke with my left hand mainly as a guide for the path. He belives I am "mixing my wires".
If my left hand was dominant, I couldnt lag to save my live.
Since you ask, I have to say that the hands are generally the enemy to touch and also to square delivery of the putter face thru impact. In my approach, the hands are really "dead" in the sense of inactive, inert, not used to power the stroke or even to guide the stroke. Gravity alone sets the free-fall tempo of the stroke I teach, and all my hands "do" is keep up with the gravity-sponsored free-fall of the putter from the top of the backstroke position down to the bottom of the stroke where my shoulders level out, and then I lift the lead shoulder up and back in a vertical plane so that the shoulder pulls the whole left arm and hand and putter up as a unit.
The shoulder action is powered by the midrift and lower back muscles, but even these muscles are unused until the bottom of the stroke. That is, the gut muscles put the putter back from the ball to the top of the backstroke (the whole triangle is "tipped" back by moving the lead shoulder down and back; from the top of the backstroke, I just relax the gut muscles and the shoulder frame drops back to level beneath a fixed pivot at the base of the neck in a smoothly accelerating manner due to gravity; as the putter head bottoms out in the middle of the stance beneath the pivot, my hands have been riding the putter down (not pulling or using any voluntary muscle activity), so that the handle of the putter stays the same inside my grip and my hands "do" and "feel" nothing changing during the stroke; once gravity delivers the putter head in an accelerating way to its peak speed right at the bottom of the stroke, I start my left shoulder up and back in a manner that respects the now decelerating pattern of the upstroke as the putter coasts to the top of the follow-thru, with the intention of the shoulder action maintaining the feeling in the hands and arms of "nothing changing." The arms are not used, so the arms stay "connected" to the upper torso the same all the time and the arm pits stay closed (in a relaxed way, not clinched). Thus, "touch" or distance control comes from the gravity timing and the backstroke length. The backstroke is set instinctively by the cerebellum based upon my targeting of the line from ball to target, and my hands never have any role to play in this.
It has been my constant experience that using a dominant hand of any sort is bad for the stroke. It is a poor way to get distance control as it uses the unsuited parts of the brain for this job, and the human voluntary effort is always inferior in consistency and accuracy to what the physics of gravity always does exactly in the same manner. Moreover, the accuracy of the falling of the putter head square back to the bottom of the stroke and thence up into the back of the ball is also best done merely by the movement of the shoulder, and not at all by moving the hands.
Accordingly, I make every effort to avoid thinking about the hands as having a role in the stroke, and specifically think only of the ball of the lead shoulder socket right before I move this socket down and back and then up and thru. If the shoulder action is performed correctly, the putter is merely an extension of the shoulder motion, and the putt sends the ball away perfectly straight. While it may be true that there are some special folk about who have a very nice handsy stroke, I'm not sure they would be as deadly and consistent as a sound shoulder stroke with dead hands over the course of a year. And frankly I doubt that a handsy stroke really has as good a diastance control as comes with a gravity-governed tempo and cerebellum control of setting the backstroke.
So, yes, golfers generally "have" a dominant hand. But so what. The issue is whether the dominance applies to putting. If you study hand dominance, you will find that there are a number of different forms of hand dominance -- hand dominance for opening doors depends on how society chooses to arrange the door knobs, for example. Just ask a lefty about that! The same person may be right-handed for signing their name, but left-handed for clapping; left-handed for holding the phone to the ear, and so forth.
I usually find that it is advisable to show right-handed golfers how NOT to let their right hand spoil the stroke. "Sticking" the right hand "out there" across the torso is very much like reaching to shake hands with someone standing just off your left side -- the forearm bones roll as the hand positions itself vertically and extends to grasp the others hand and shake it. This arm-hand action is a pull stroke, whereas a good putting stroke is a "slap to the sky" of the right hand and a "slap to the sky" of the back of the left hand (right handed golfer), but the deliberate moving of the hands is only a result of moving the shoulders, or more properly sending the lead shoulder socket down and back, then up to level, then up more and back, so that the hands aren't specifically being moved. The trick to redcing the pull (hand shake) tendency of the right hand (aside from thinking only of moving the shoulder) is to loosen the thumb of the right hand on the grip so the "precision grip" of thumb pressing at side of index finger in tool use is deliberately disabled. Douing this, plus thinking only of moving the lead shoulder pretty much overrides the engrained "pull" tendency the dominant hand injects.
So much for the stroke. For AIMING the putter face, the non-dominant hand is usually more accurate in positioning the putter face so that it aims sideways most directly at the target. This is because the non-dominant side of the body is controlled by the non-dominant side of the brain, and the non-dominant side of the brain is best at spatial awareness and orientation. For a right-handed golfer, this usually means that the left hand is better at placing and aiming the putter in setting up to the ball.
So I completely have gotten away from notions of hand dominance for powering or guiding the stroke, and find that touch and feel is really a stable tempo (gravity is the MOST stable tempo available) plus targeting that allows the cerebellum to set the length of the backstroke.