Should I notify competing researchers that I am working on the same problem?
Norms on this probably vary between fields (not to mention between individuals in a given field). In my corner of theoretical physics it's not uncommon for people who know that another group is working on a similar problem to check in with each other on the status of the project and, if they are at similar stages, to coordinate posting papers on arXiv simultaneously so that no one scoops the other. This also has the advantage of helping to publicize the result--if someone skimming new articles notices that two or even three groups have put out papers on the same topic at the same time, it can attract more attention than the papers may have if they were spread out.

It has also been my experience that even when two groups are doing roughly the same thing at the same time, and even when the results have some overlap, they typically end up with different perspectives and some amount of complementarity. Sometimes the results even have no direct overlap but, taken together, build a stronger case. So I would generally not stop working on something just because someone else is, even if I know or suspect that they're farther along.