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If you give someone the host, it becomes their personal property; THEY get to decide what to do with it.

I don't think it's that simple. How is there a conveyance of title?

In the Catholic Church a host is given to a lay person for him to consume or, in extraordinary cases, for him to convey to another person for THAT person to consume. In really rare cases it may be given to be kept in a suitable place for adoration or exhibition.

It would be an interesting case to hear, but I don't think it's a slam dunk.

I'd not assume that the Host became the unbeliever's "personal propertry." I'd argue that he obtained the host on the basis of falsehood, inasmuch as he pretended to be a believer and a communicant, for the priest or EM wouldn't have given him the host otherwise: and he, and and everybody else, knew that. It is a kind of fraud.

An analogy might be a funeral of a soldier in which the mourners are given some token (a small American flag, a photograph of the soldier) at the gravesite by the soldier's family. If a pretended mourner came and were given these objects, and then proceeded to burn them, smear with filth, or otherwise dishonor them--- saying that he was given the tokens, and the soldier's family had thereby relinquished control--- he should not expect to be held harmless, even legally.