I don't recall ever seeing anyone say getting vaccinated makes you test positive for HIV. I've heard people saying the effect of the vaccines on your immune system are LIKE having AIDS, meaning it trashes your immune system. But that wouldn't cause you to test positive for a different virus.
Vaccine induced shedding only happens when a vaccine contains a weakened, live virus, that happens to manage to reproduce, and essentially gives you whatever that vaccine was intended to prevent. The worst you would get from these shots might be the spike proteins in the shots, which could be bad, but I haven't seen any studies about either the escape of the vaxx components from one body, or the subsequent interaction with another person.
The sex thing, I still don't know. I've heard that one, but haven't seen anything I'd call reliable.
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