In December 2015, the Board of County Commissioners amended the County Land Use and Development Code to allow short-term rentals in all standard zone districts such as Rural or Residential. This allowance does not include Planned Unit Developments districts unless the individual PUD zoning permits such use. You can find the PUD Guides here.
It's always good to contact the Planning Division if you have questions about permitted use in your location. You can contact the Community Development Department at 970.945.8212.