The DoTs on the enemy show up as red, actually.
However, be careful, you won't see a million red numbers if the enemy has 30 DoTs (i've seen enemies with 30)

All the DoTs tick synchronically, timed by the server, so they all tick together. And the red number is the sum of all the DoTs going off at the same time. So say all you deal a grand total of 500/tick, your Dragoon friend has like 200/tick and your white mage and black mage there also have about 120/tick each. Once the DoTs tick you will see a single 920 red tick