Both arrangements exist, which is why you got a confusing answer. Some travel healthcare workers take agency-provided housing, where the staffing agency arranges the unit and the nurse never shops. Many others take a housing stipend instead, a cash allowance they keep whatever they don't spend, and those people absolutely do go looking for a place themselves. Stipend takers are the mid-term landlord's main market, and they're searching furnished housing sites, hospital-area Facebook groups, and word of mouth from the nurse who finished the contract before them.
So posting a listing and waiting is not nothing. It's just slow and passive, and the operators who stay full add two things to it. The first is referrals: ask every departing tenant whether anyone on their unit is rotating in, because contract cycles run in batches and one nurse often knows three more arriving the same month. The second is the housing coordinator or relocation contact at the hospital or the big local employers, who often keep an informal list of nearby furnished options to hand to new arrivals. Getting on that list is a phone call and a follow-up, and it costs nothing.
Write down the hospital's contract start dates if you can learn them. Travel contracts commonly start in clusters, and if your unit comes empty two weeks after the cluster starts you'll sit for six weeks waiting for the next one. Aligning your lease end dates to land just before a start window matters more to your occupancy than the photos in your listing do.
Don't furnish that duplex side until you've made a few of those calls and confirmed there's stipend demand rather than only agency-placed housing.