I'm not sure what you mean by begin and end. The negotiations begin when you make an offer and end when you come to terms. That could take hours, days or sometimes even weeks. There may be additional negotiations over inspection items later, after you come to terms.
It's unlikely you can actually hire the listing agent. In Washington if you make an offer through the listing agent it is more likely that the agent would still just represent the seller, and you would be unrepresented. Dual agency is rare. Also, the standard listing form doesn't call for a reduced commission to the seller in either instance, but it is possible that there is such an agreement. If the listing agent is a "Realtor" that would need to be disclosed in the listing.