You're not getting clear answers because there aren't any.
Competition/cooperation with local ski areas will depend on where you are talking about. In Colorado (where you are located), most ski resorts are on USFS land. Technically, there are rules and permits that apply for doing business on USFS land. Many one person operations simply ignore these with impunity. Whether resorts are on USFS land or not, they generally have exclusive rights to conduct business on the property they operate on and usually prefer to have all instruction on their mountain conducted through their ski school. It is possible to operate either with their permission (under their USFS permits or your own) or to simply operate under their radar. In my area (Mid Atlantic) the local resorts tolerate private coaching conducted through local ski clubs because of the group business the clubs bring to the resorts. The resorts also tolerate a few private coaches who keep their operations very distinct from the resort and very "quiet" while on the resort. You might find this thread about Steamboat to be interesting reading.
Liability insurance is not cheap (e.g. $60-100/day) and not easy to find. There are 3 basic ways to get customers: network (i.e. find parents with money, build relationships with industry insiders), build a community (e.g. write a book, make a video, run a website), build a rep (e.g. coach successfull freestyle competitors).