Most of the times libraries do not create their own digital reference service. They themselves use a service called QuestionPoint.
QuestionPoint is a Web-based virtual reference service and administrative tools for libraries of all sizes. QuestionPoint reference management service provides libraries with tools to interact with users in multiple ways, using both chat and email. Added to a library website, QuestionPoint gives users access to quality online reference assistance. Behind the scenes, QuestionPoint administrative functions help libraries manage all their reference transactions effectively. QuestionPoint uses a web-based chat tool with co-browsing capability, coupled with the email reference component, enable seamless integration of chat, follow up and referral, as well as one-stop reporting tools for all types of reference services. In addition, libraries may opt to participate in the 24/7 Reference Cooperative to provide live around-the-clock reference service to their community.
QuestionPoint is a collaborative project of the Library of Congress and the Online Computer Library Center (OCLC).
Ref: http://www.questionpoint.org/