Private (trespass fees) does seem to be the way to go.
Unless you know of public land that holds elk. Elk are easily visible on private land in stream valleys because it's open. That doesn't necessarily mean there are none on the surrounding public lands. Every year I run into guys at the trailhead who tell me they've seen no elk on public land and heard no bugles. Then I hike in a ways and get serenaded by multiple bulls.
