In The Basic Problems of Phenomenology, page 92, Heidegger writes:
If we compare it with the Kantian thesis, the Thomistic thesis
says—indeed, in agreement with Kant—that existence, there-being,
actuality, is not a real predicate; it does not belong to the res of a
thing but is nevertheless a res that is added on to the essentia [concept]. By
means of his interpretation, on the other hand, Kant wishes to avoid
conceiving of actuality, existence, itself as a res; he does this by
interpreting existence as relation to the cognitive faculty, hence
treating perception as position.
This is set out in Kant's Critique of Pure Reason A598/B626 (at page centre):
Being is evidently not a real predicate, that is, a conception of
something which is added to the conception of some other thing. It is
merely the positing of a thing, or of certain determinations in it.
Logically, it is merely the copula of a judgement.
That is, in making the judgement "people are in that room" they thereby exist in one's mind, which is the only reality one can be absolutely certain of. The "are" constitutes the copula of a concept (people) plus predicate (in the room) brought to existence by the judgement. Concept plus predicates is the constitution of a thing since Aristotle. This Kantian form of existence applies to things observed, not the observer's mind.
OP: Does "in the room", "in the school", etc all imply the existence?
So no, they are predicates which would need to be joined to a concept in judgement for existence to be actualised.
It follows that "people are not in that room" asserts where people are not and does suggest that people might be elsewhere. The statement "unicorns are not in that room" strikes one differently and prompts the objection that the concept is not realistic.