Some customers want to know that their support system is nearby. They want to walk through the facility and meet the people. They want to know that if there's a problem, they can look into the eyes of the person who is responsible for fixing the problem, and get the answers they need. This becomes a significant challenge when dealing with offshore service providers whose facilities are thousands of miles away.
Talk to your major customers before outsourcing a key function overseas. Some customers may cancel their contracts if service is outsourced offshore.
What can you do to help both internal and external customers "meet" their providers?
Plan for a representative from the provider to visit your site in person. Face-to-face contact can't be replaced. This will help the provider to know for whom they're providing service, as well as giving them an opportunity to meet more of the staff of your organization.
If possible, you and your immediate team should visit the provider to help establish a relationship. Meet as many people as possible during this trip.
If you cannot visit the provider personally, ask for a video of the provider's facility, along with some personal commentary by the principle employees. This might help your organization and your customers to feel more comfortable.
If your customers are external, provide a local "neck to choke." Customers need someone who takes full responsibility for the outsourcing provider's quality of service. (After all, your decision to outsource is not your customer's problem.)