Demi isn't an Island artist yet. She's under HWR and Safehouse which means HWR will handle promo like previously stated and the fact that she's under Safehouse too and they have a deal with Island means she will have Island intefiring as well, but not directly. For now, until HWR contract is up and then I think she might end up being an Island artist as well, but she's just a HWR/Safehouse artist now. The Island association comes with the deal they have with Safehouse, not because they signed her yet.
And yes, like one of you stated, Republic manages promo and radio distribution for Island artists.
Nick founded Safehouse last year and got his album distributed by that label, so Demi's album will probably be distributed by Safehouse too.