A simple method to experiment with head weight, putter balance, etc. is to check and see if your putter head has enough iron content to hold a magnet. Most will except for aluminum heads. I use this method because you can widely vary the amount and placement of weight in seconds right on the putting green. It simply requires a handful of the very small little round magnets or even better rectangular if you can find them. You can use magnets on the putter head and a magnet strip with or without small magnets on a steel shaft if you are seeking to weigh the shaft which some players do.
Btw, this is also an excellent way to practice to develop face center contact with your putter. I use two rectangular magnets and position them on the face separated with just enough room for the golf ball. If you are off you will contact one of the little magnets. Depending on time, I do this occasionally before a round, but I do pull them and practice without them for a few minutes before the first tee.
Adding weight down the shaft on a putter will generally affect the "center" of the face. However, I worked with some very good players that wanted the "center" of the face of their putter moved closer to the shaft. I never had a good player or any player that wanted the "center" moved toward the toe, but I suspect they are some that do.
Another interesting experiment is to weight the shaft to balance it. This is usually done with sand and a mix of light weight plastic balls, if the player wants the weight balanced through the shaft. Once the desired feel is established you then need to use something to fix the weight as you violate the rules of golf to have moving weight. What I have noticed over the years is that players that use their shoulders to stroke tend to be those wanting the heavy or weighted shaft. The player that used more hands and wrists seeking to keep the upper body still will generally want the weight in the putter head.
Rarely ever discussed is how many of the best players in the world will make alterations to their putters. Just like you can buy a club from an OEM thinking I am playing what "latest greatest" tour player is playing. Well it may look the same, have his name on it, etc., but is not the same. They have a whole team customizing their irons, drivers, etc. to personal preference. The alterations to putters is at least equal and can vary to variety of grass and speed of greens from one week to another.
Bubba stated his most recent putter and the one he used in the recent match play event made by Ping was actually two putter designs welded together to fit his eye. He also said he now leaves it up to the Ping engineers to tweak his putter for him. You can be sure they have worked with him for many hours over a period of time. This is one of the first times I have seen a tour player actually admit (indirectly) what may feel best to him may not be what he actually needs. You may find the same if you experiment and then keep a record of what actually performs best for you.