Frodo's necklace isn't invisible; therefore, we can assume that it does not work unless worn on a body part specifically, as he is wearing it while it is on the necklace but he is not invisible.
The real question is what body parts would work. Fingers obviously do, but would toes? Cockring? Nose piercing? Could you say that the ring being in his butt counts as wearing it? If someone can go outside wearing nothing but a buttplug, then there is some president.
Can only buttplugs be worn, or can anything be worn in the butt? Is a ring a buttplug? Cash or credit?