User fields, such as assignee can be automatically mapped in JIRA Jira and TFS / VisualStudio Visual Studio Team Services, provided that both JIRA Jira and TFS / VSTS server are connected to the same Active Directory instance and therefore share their user account data.
For instructions how to connect JIRA Jira to Active Directory please refer to https://confluence.atlassian.com/display/JIRAJira/Connecting+to+an+LDAP+Directory.
To enable automatic mapping for "Assignee" field please click the "Values" link for "Assignee" mapping and select Mapping Mode "Copy Value (Attempt Type Conversion)". Please see the screenshots below for details.
