The part of the axe with which I am concerned here is what is labeled in the above picture as the Axe Blade/Cheek, and Cutting Edge Curvature/Sharpening Bevel.
A good multi purpose axe will have both a thin edge curvature, and thin cheeks/overall blade. Not all axes need to be ground that way. There are many tasks which can benefit from a more robust tool, but I think a thin blade with a narrow edge curvature makes for a good overall wood cutting blade.
Some axes, fall outside of that description. Some have thin cheeks, but the edge curvature is too wide/thick, while others are thick overall.
In the above picture you can see three different axe grinds. Open the picture for a larger image. The first is what I would consider good. The cheeks/overall blade is thin and the edge curvature is narrow.
The second, shows a grind that is overall too thick. As you can see from the picture, if you attempted to grind down the edge of the blade, because the whole head is so thick, you would have to grind down the whole head. I would not bother with such an axe. Such a re-profiling would take a huge amount of work.
The third grind is the type that I would not consider ideal, but is very workable. The cheeks/overall blade is thin, but the edge curvature itself is wide/thick. As you can see, it is possible to thin out the edge curvature without effecting the rest of the axe head.
The thinning out process involves filing down the thick part of the edge.
It is similar for the way you would sharpen a convex edge with a file, but instead working the whole convex curvature of the blade, you only work the parts you want the thin out. Eventually you will work it down to a level where you get the curvature you want and can transition to sharpening.
For this kind of job I would recommend using a file. I’ve been using a 200 grit 8 inch file with good results. Depending on how much metal you have to remove, you may want to get a more aggressive file. I would not recommend using a grinder because the heat form the grinding process can ruin the temper of the edge.
All modern axe heads are workable in this manner. Some older ones, from prior to the 1950s may be too hard for some files. You will need to find an extra hard file in order to work on them. Most however, should not be a problem.