Why not reformulate the question to something so that it becomes well posed or meaningful if you know it can be done ? Specially when it becomes a valid well posed question, something which is obviously much more interesting than which was originally posed ?

But that requires me putting on my wizard's hat and dragging out my crystal ball.

If the question is already clear enough that it is obvious what the person should be asking about, but just is missing some technical points, then I wouldn't call the question "ill-posed" to start. The questions I consider "ill-posed" are those which one has to pick at the Original Poster's brain to figure out what is meant. (Note: ill-posed is not the same as easy for someone in the field!) The best way for the question poster to gather good will in the situation you are asking about is, I think, for the OP to include all motivation (especially those from his field of specialty) on why he is led to consider the question. This will, on the one hand, let the community know that this is not an idle question, but one involving real research (just not in the OP's field of specialty); on the other, allow people to better assess the appropriate way to interpret and answer the question if it is not completely well-posed.

In fact, from what I've seen on MO the people are very forgiving: if the question is ill-posed as stated, there will be comments prodding the OP to clarify the necessary parts. But for fear of mis-interpretting the OP and making the question something completely different, I would advice against a general policy of editing the question to "make it better" in the case where the original meaning is not clear.

Presumeably you've seen

http://tea.mathoverflow.net/discussion/704/diploma-thesis/

and

http://mathoverflow.net/questions/43755/when-should-you-and-should-you-not-share-your-mathematical-ideas

if not, you should give them a look-see. For what it's worth, MO is a public forum. So while sharing your ideas on MO can bring to you some competition, it is slightly harder for someone else to "steal your credit".

That said, one of the best things to do in this context, which you already thought about, is to find another young PhD/PostDoc who is an expert in that other field and start a collaboration. You'll both benefit from it. You may even consider writing a short letter to the author of that paper you described to ask whether he will consider a collaboration, and if not, whether he had any students he can suggest you to contact. (I don't think asking on MO would be appropriate strictly speaking, but maybe you can ask on Meta? Considering the recent thread on Canadian mathematicians, I don't think that is too much of a stretch.)

You can post. But it will hardly be useful for you if it gets closed rapidly as too-localised/off-topic or worse spam.

MO expects certain kinds of questions. If you willfully ignore that expectation, you may garner ill-will towards you. Just saying...