Once you have spotted a mathematical problem that (presumably) fits your degree of expertise, whether you are a phd student or an established professor, you have to deal with the following non mathematical problems:
- How to know if somebody else in the world is already working (or has already been working) on the same problem?
If the other guy has already completed a certain amount of (say, not yet published) work on that specific topic, knowing this would help you to avoid waisting time to try to re-do something that has already been done (at least with the same methods).
On the other hand, if the problem is broad enough, knowing of some other's interest in the same context would also be fruitful because you'd may have somebody with whom to talk and to whom to ask questions, without overlapping the specific research goals. Or you may even find a collaborator.
- In some cases the very choice of an interesting specific problem can be a nontrivial task by itself. So, in case you want to ask around if some previous/present effort has/is been made in that specific direction or related ones, should you worry about the possibility that somebody with a higher degree of expertise would just "take your problem" and solve it faster than you would do?
I'd expect the obvious answers, such as "have a look to mathscinet/arxiv" or "search the literature" or "talk to people (your advisor if you're phd)", to be enriched -if possible- by some more elaborate viewpoint or more specific suggestion.