I honestly do not know for which individual species this method has been used, and in particular whether it has been used for Puma concolor, but a general method that is frequently used for population estimates is "mark and recapture"
In this technique, you enter an area, trap as many of the target species as you can, and mark them, such as with a collar or leg band, keeping a careful count of how many have been marked. Then you come back a while later and repeat the process, counting how many of the newly trapped animals were already marked.
Now, your population estimate is basically a proportions problem from high school math class: M2/C2 = M1/P where M2 is count of marked animals in the second sample, M1 is count of marked animals in the first sample, C2 is the total count in the second sample, and P is the total population. You know the first three of those, and just need to solve for the fourth.
Update: I found this academic paper which indicates at least one study in Montana that has used a modified form of mark-and-recapture on Puma concolor.