Basically, you want to learn HOW TO CHECK WHERE IN FACT THE PUTTER FACE AIMS, and then PREDICT where you think it aims by identifying a spot on the ground or a spot along the base of a wall where you judge the putter face to aim, and ONLY THEN turn on the laser to find out whether you are any good at this skill. If the laser shows up where you thought it would, you're good; and if to the left or right, then you need to keep working on getting the skill down.
The above learning / practice protocol is opposed to the usual one of just setting up and then turning on the laser and then pointing the laser beam at the target and then sort of trying to absorb something about the "look and feel" of good aim.
Instead, the skill to learn is how to stand at address beside the ball and aimed putter face and run your line of sight like a laser beam straight sideways along the ground on the same line the putter face aims down. This requires three components of body positioning-posture-movement. First, aim the eyes straight perpendicularly out of the face; second, align the line of the throat from base of neck to point of chin with the leading edge of the putter or at least set it parallel to the putter face leading edge; and third, after bending the face down with the gaze held steadily straight out until the back of the ball and the putter sweetspot show up, rotate the head and face to send the line of sight straight sideways like spinning an apple on a stick, the stick being the axis of rotation of the center of the neck out the top of the head. If you also use only the rear eye (close the eye nearest the target) and are sensible about the "peak" where the ascending line of the nose meets the descending line of the eyebrow, the spot on the ground along the line sideways where the putter face actually points will occupy the inside of this peak, as shown in the following drawing:
Once you learn this skill, then light up the laser to see how accurately you have predicted where the putter face actually aims. You will probably find that of the three components, setting the eye gaze straight and setting the throat line to the leading edge of the putter face are easier to get right than the head turn. That is because humans turn their heads all the time while standing up, and this sends the chin at the shoulder, whereas in the bent-over posture at address for putting, the chin needs to stay an equal distance out from the line of the shoulder frame the whole time the head is rotating. To practice this particular aspect, set up next to a line on the floor, make a tiny hole with your rear-side fist and use it like a small telescope held perpendicularly to your rear eye set aiming straight out of the face, bend the face and hand together until the line on the floor shows up inside the timy telescope, and then turn the face on a rotating head SO THAT the line on the floor stays inside the fist telescope as the top of the head where the button on a cap would be simply spins in place without sratching backwards out of position.
Clyde Melancon makes a very neat device for $22.50 called the "GTA Straight Gaze Laser Glasses" with a laser mounted along the sidepiece of a glasses frame, so that a laser line projects where a straight-out gaze should aim, and THEN the laser line will run straight away from the putter face ONLY IF the head turn is done correctly.
Putting Coach and Theorist
Golf's most advanced and comprehensive putting instruction.