First, 人だ does not actually mean "a person exists". That would actually be 人がいる. 人だ actually says "(it) is a person" (that is, it is describing some other thing as being a person).
だ/です ("is") is a bit special in Japanese in that unlike verbs (or unlike other verbs, depending on whether you consider だ itself to actually be a verb or not (opinions differ)), it always takes an argument immediately before it (without any particle). This argument is generally either a noun or a な-adjective.
So in this sentence, 人 is a noun, and だ is the copula (the verb/grammatical construct which expresses equivalence), which always comes immediately following the noun/な-adjective which it is saying something is.
As far as "part of speech" it could arguably be considered to be the "object" of the copula (more formally referred to as a "predicate"), but unlike other verbs the copula just does not use を to indicate its object, but instead just immediately follows it. Alternately you could say that it has a part of speech which doesn't have an equivalent term in English (I'm not sure if there is a Japanese term for it or not)..