The only time a 4473 is used is when purchases are made through a FFL. For some states, they require all private sales to go through a FFL. The FFL must hold onto all 4473's for as long as they are in business. If a FFL closes for any reason, then all 4473's are to be turned into the ATF. For states that a required background check is handled by the state, they will have their own paperwork and procedures for private sales.
To answer your question. Missouri does not require any background check for private sales. You just have to follow state and federal law. And state laws here mirror federal laws. Both seller and buyer must be residents of Missouri for all private firearm sales.