Johnson County publishes the roster of permitted disposal sites in its area, and two of them are publicly run.
The nearest public transfer station is the City of Olathe Solid Waste Transfer Station at 1681 S Valley Road, Olathe, KS 66061, on 913-971-9311. It is a municipal facility, listed by the County among the county's permitted transfer stations.
For hazardous material, the Johnson County Department of Health and Environment household hazardous waste facility sits at 11231 Mastin Street, Overland Park, KS 66210, on 913-715-6907. It operates by appointment only, and it is a hazardous waste route rather than a place to send a container, so please do not plan a remodel around it.
Construction and demolition material from Leawood goes to a permitted construction and demolition landfill instead. The County names two in its own listing that sit within a short drive: one at 7100 W. 167th Street in Overland Park and one at 23200 W. 159th Street in Olathe. Both are privately operated, so each sets its own gate charge and its own acceptance rules.
One other thing worth knowing. Johnson County's health and environment department licenses the residential waste and recycling haulers that collect in the county, which is a household collection system rather than a construction one, so it is not the obstacle it might look like for a project container. Your account manager names the destination and its charges before delivery.