Did they both just come about separately, and it's a freak coincidence, or did one ultimately influence another? If there is a shared significance, what is it?
It's not so much a coincidence as it is a generalization. The romanization of the Mandarin pronunciation for the creature which we know as dragon is spelled "lóng", but without the tone marks it becomes simply "long". This makes it cumbersome and awkward as the word already has a meaning in the English language is commonly used in daily life.
Calling the 龍 a dragon is a generalization rather than coincidence seeing as how there are mythical beasts of reptilian nature in numerous cultures. However the role occupied by the Chinese dragon is extremely different from European dragon.
The dragon in China has long been seen as a harbinger of luck and prosperity, while in Europe the dragon is seen as something to be hated and feared, the devil incarnate. It is seen as a source of power, but to be defeated and overcome.
The 龍 is more akin to a unicorn than dragon, while the European dragon is rather more like the Nian than lóng, if that makes sense.
http://en.wikipedia.org/wiki/Nian
Well actually the Nian could also be seen as analogous to Grendel, but I don't know enough about Chinese mythology to provide another example. It's Chinese name is 年獸, which just means Year Beast.
You'll find more discussions on dragons here. In short, the only connection is that they share the English name "dragon" and are sometines depicted with reptilian features. Their symbology, meaning, and general depictions have few similarities.