How can I tell if stains are caused by a ceiling that is drying out, or if there is still a leak?
We had a leak going into our ceiling. Luckily, as a wood turner, I already own a moisture meter. This can test the moisture content of wood (but other similar density substances too) merely by pressing a button. That meter worked very well for my problem. Of course, if you do buy one, I don't expect you will find many other uses for it than catching an occasional leak, but it can still be cheaper to buy one than to pay someone to make a simple reading for you. Just store the meter after you are done with it with the battery removed.

It will take several weeks or even months for a ceiling to dry out. This depends on how enclosed the area is, and whether there is still water getting in. Readings made over time should be decreasing. If not, there is still a problem. The moisture content of even dry wood will be around 8%. Check other walls in your house, because this number will vary by where you live, and what time of year it is.

There are several styles of moisture meters. One relies on pins that must be inserted into the material, testing electrical conductivity between the points. As such, it will leave small holes. Better are the styles that do not leave a hole, but they are more expensive. Note that the cheaper models use a probe. This probe will have pins to stick in the wood.

By the way, while the ceiling is still damp, stains WILL continue to spread by capillary action. The water will continue to wick the stain further away. 2 months later is a long time for new stains to appear, however, old stains can bleed through new paint. You may wish to use a product like Kilz to seal in those stains. But wait until it is surely dry before you paint. This is where the meter will help you out.