There's been a couple of questions with titles like "bibliography problem" recently. Is it within the remit of those with editing privileges to change the title of the question to reflect the actual novelty of the question asked?
In general, yes, you should edit without hesitation.
Titles are particularly important since they are the signposts that will invite (or scare away) fellow community members -- and the greater internet -- to visit and participate.
I think it's better to assume up front that you are working collaboratively with the post author to improve the post, rather than asking permission in advance.
IMHO: If the question is actually more specific than the title makes it appear than it is OK to have that changed. It might be better to first comment on it and ask the OP to adjust it. It is still possible you misunderstand the question. However, users new to (La)TeX might not be able to scope the issue they have correctly. Sometimes new TeX.SX users have also problems with editing their posts. If the title is really generic and not very meaningful like "bibliography problem" or even "What is causing the following problem?", than I personally would (and do) change it very quickly to keep the standard of the site high. Waiting for the OP to edit it could take days in some cases.
There is also the other directions: Some questions are about some solution for/using a specific thing which isn't really required. This means neither the problem nor the solution is specific for it. For example, I recently changed the title of a question asking about a list of to-be-ignored auxiliary files from specifically referencing GIT (a version control software) to version control (VC) in general. The way that list is provided to the software may be different but is not part of the question (and would be off-topic anyway). However the same list can be used for many different VC software products. So if someone is looking for that list for, say Subversion, the answers will also help and so there is no reason to have to have a Git specific title.
If the goal is to improve question titles generally, then I think adding a comment asking the questioner to make the title more specific might be a better tactic. It would raise awareness not only to the person asking the question, but to others who might read the question. A direct edit doesn't achieve this.