Identify these two large, colorful dragonflies in Taiwan?
Looks like this species is Orthetrum triangulare commonly known as blue-tailed forest hawk. It is an Asian fresh water dragon fly species.
As you mentioned that the dragonfly had a bluish hue, it became clear.

The other one looks like Neurothemis ramburri, commonly known as red perch Reference

Neurothemis is a genus of dragonflies in the family Libellulidae. Most Neurothemis species are red in color.
Neurothemis ramburii (Brauer, 1866) is distributed from Sundaland to New Guinea. There are records from Peninsular Malaysia, Taiwan, the Andaman Islands, Sumatra, Java, Borneo (Sarawak, Sabah, Brunei and Kalimantan), the lesser Sunda islands, the Philippines, the Bismarck islands and New Guinea. It is a widely distributed and common species that thrives in disturbed habitats.