Emptiness of person is the emptiness of a subjective self, agent (perceiver/controller/doer/thinker), soul, being, within or apart from the five aggregates. One sees that the framework that "a seer is seeing the seen" is delusional, I.e.in seeing always just the experience of the seen, colours, shapes and forms without a seer. One realizes that a subjective self is illusory.
Emptiness of phenomena means each of the five aggregates does not have objective existence. There is no formness in form, no substance of thought, etc. So they are appearing but empty, illusory like a magician's trick, like a mirage, a dream.
Here is an excerpt from a Buddhist glossary site on the definition of twofold Emptiness:
Two emptinesses (二空) include (1) emptiness of self, the ātman, the soul, in a person composed of the five aggregates, constantly changing with causes and conditions; and (2) emptiness of selves in all dharmas—each of the five aggregates, each of the twelve fields, and each of the eighteen spheres, as well as everything else with no independent existence. No-self in any dharma implies no-self in a person, but the latter is separated out in the first category. Realization of the emptiness of self in a person will lead to attainment of Arhatship or Pratyekabuddhahood. Bodhisattvas who have realized both emptinesses ascend to the First Ground on their Way to Buddhahood.