The short answer is this:
The sending church should take responsibility for guiding and shepherding and supporting their missionary in every way, from initial candidacy to retirement.
The supporting church may have as little relationship as functioning like a paymaster or as significant a relationship and ownership of their ministry as the sending church, without having "the buck stops here" responsibly.
One pastor, speaking to one of the supported missionaries of the church, said, "We may not be your sending church; but we want to become your favorite church." Such warmth and dedication bodes well for that churches relationships with all of their missionaries.
A supporting church needs to respect the role of the sending church and, if necessary, encourage the sending church to step up their responsibilities. It is normal in the application process for support that the supporting church would ask the prospective supported missionary who is their sending church and who is the primary contact person in that church responsible for oversight of their ministry on the field
Likewise, the sending church should find out the names and contact persons of other supporting churches of their missionary. There are a number of times when knowing that information can be useful. E.g.:
- coordinating STM teams to their field ministry
- making urgent prayer requests known on their behalf
- coordinating promotion and development for special fundraising needs or projects
- enlisting resources and help for special needs for counseling, housing, transportation, health issues, etc.
Certain crisis situations may require that the supporters back home act quickly and in concert with one another in order to respond best to the crisis at hand. Prior knowledge of contact information and a respectful acknowledgment of sending in supporting roles can be of great comfort and help if and when a crisis really happens.